It needs at least the same signal your GPS watch needs. The better signal you can get it, the better time solution it'll get, and the better holdover timing it'll keep. Here's why:
Imprecision in GPS (GNSS in general) solutions comes from a number of sources: Orbital errors in the satellite ephemerides, atmospheric distortions to the signal, multipath at the receiver antenna, and a bunch of minor stuff we're not gonna worry about. Those are the big three.
Orbital errors, you can't do anything about. Post-processing allows _position_ solutions to be corrected once the errors are known (a process that takes hours or days after the fact, to work back where the satellites must've actually been) in a process called Precise Point Solution, but that doesn't apply to [this kind of] timing.
Atmospheric errors you can kindof do something about. Signals from satellites lower to the horizon take a longer path through the atmosphere, so they tend to have wackier pseudoranges. Most timing-grade receivers have a default "elevation mask" of 15 or 20 degrees above the horizon, to simply not even bother with these crap signals, and derive timing only from what's left. (There are of course atmospheric models and corrections to apply, but these are applied to everything. Still, start with a cleaner signal and you'll end up with a cleaner result.) This means you need clear sky view to the middle of the sky, in order to raise the chances of an adequate number of satellites passing the elevation mask.
Multipath is huge. The reason survey-grade antennas have these enormous choke-ring structures around them is to eliminate multipath reflections off the surface of the ground below. Even cheap little antennas try to be more sensitive to signals from above than below, to limited success. (Especially if they're wrist-worn and thus not orientation-controlled. Ugh.) Any situation where the antenna is subject to multipath will have demonstrably worse precision as a result. So, most timing antennae are considerably elevated (gives the reflections an opportunity to fall apart), shielded (choke-ring mounts), or both. Cellular sites usually go with the elevation option since there's already a giant lightning-rod nearby, elevating the GPS antenna doesn't come with additional risk for the receiver.
Worse GPS reception means the receiver has a noisier signal to start with, and can't discipline its local oscillator as tightly. Even if you don't care about nanoseconds in the final application, you probably do care about _holdover performance_. And if your LO can be disciplined with nanosecond-accurate signals in the first place, it'll hold millisecond-accurate timing for days or weeks in a GPS outage. If the disciplining process is crap, the holdover will also be crap.
Doubtful of the last paragraph: What kind of clock oscillator 'learns' to keep better time by being adjusted from a better source? A normal crystal doesn't have any memory effect AFAIK.
In a typical real life scenario where the GPS / antenna malfunctions, your server clock will just slowly.. drift.. I.e. the crystal runs a bit too fast or a bit too slow.
The last paragraph is what the FB's timecard (and essentially any other rubidium based frequency normal) does. You use the GNSS frequency output as a reference for PLL that does fine tuning (on the order of tens of ppm) of the rubidium oscilator (or even normal crystal oscilator). The reference oscilator modules usually have some kond of input for this kind of fine tuning (in xtal case this analog input typically controls biasing of varactor that introduces parasitic capacitance to the crystal, in rubidium case it is usually done digitally and it changes the division factor in the feedback path of internal PLL).
Edit: and for that matter even ntpd/chrony does this tuning in software and can compensate for imprecise and/or slowly wandering host clock generator (obviously it cannot compensate for abrubt changes and intentionally does not even try to).
For a quartz oscillator, it's typically done by holding the crystal at a constant temperature ("ovenized", "oven controlled"), and then varying its loading capacitance by manipulating the reverse bias voltage of a varactor (a diode whose junction capacitance varies in a well-defined way according to applied voltage). You connect a lots-of-bits DAC and a very stable amplifier to the varactor, and keep all those pieces inside the same oven so they're all isothermal.
Apply a very long time constant (typically on the order of 11h58m out to maybe 2 weeks) to the control loop that drives the DAC, and after a few loop-times you've got a quartz crystal whose period is precisely tuned and whose behavior will remain incredibly stable when the disciplining reference goes away. (As long as you're smart enough to notice the loss of reference and freeze the DAC value, rather than naively assuming there's an enormous difference and slamming the control loop against a limit and destroying your carefully-honed performance. Which is what pretty much everyone implementing this from scratch does at least once.)
For rubidium atomic clocks, the disciplining process is usually done by varying the magnetic field applied to the vapor cell. Since these work by measuring the energy absorbed by a specific hyperfine transition, but that energy is only well-defined at zero static magnetic field, it's necessary for the physics package to cancel out the Earth's magnetic field. This is done with a set of Helmholtz coils, but how do you know how much current to put through them? By comparing the measured frequency with an external reference of higher quality. Since Rubidium oscillators are inherently several orders of magnitude more stable than quartz, this is done with very long time-constants (since any apparent error is probably jitter on the GPS receiver's part).
Ovenized quartz GPSDOs are common at cellular tower sites, whereas rubidium finds more application at central offices. I own clocks that exploit both of these mechanisms. (The rubidium is down right now while I build a new DAC for the tuning voltage that drives the coil current amplifier.)
Imprecision in GPS (GNSS in general) solutions comes from a number of sources: Orbital errors in the satellite ephemerides, atmospheric distortions to the signal, multipath at the receiver antenna, and a bunch of minor stuff we're not gonna worry about. Those are the big three.
Orbital errors, you can't do anything about. Post-processing allows _position_ solutions to be corrected once the errors are known (a process that takes hours or days after the fact, to work back where the satellites must've actually been) in a process called Precise Point Solution, but that doesn't apply to [this kind of] timing.
Atmospheric errors you can kindof do something about. Signals from satellites lower to the horizon take a longer path through the atmosphere, so they tend to have wackier pseudoranges. Most timing-grade receivers have a default "elevation mask" of 15 or 20 degrees above the horizon, to simply not even bother with these crap signals, and derive timing only from what's left. (There are of course atmospheric models and corrections to apply, but these are applied to everything. Still, start with a cleaner signal and you'll end up with a cleaner result.) This means you need clear sky view to the middle of the sky, in order to raise the chances of an adequate number of satellites passing the elevation mask.
Multipath is huge. The reason survey-grade antennas have these enormous choke-ring structures around them is to eliminate multipath reflections off the surface of the ground below. Even cheap little antennas try to be more sensitive to signals from above than below, to limited success. (Especially if they're wrist-worn and thus not orientation-controlled. Ugh.) Any situation where the antenna is subject to multipath will have demonstrably worse precision as a result. So, most timing antennae are considerably elevated (gives the reflections an opportunity to fall apart), shielded (choke-ring mounts), or both. Cellular sites usually go with the elevation option since there's already a giant lightning-rod nearby, elevating the GPS antenna doesn't come with additional risk for the receiver.
Worse GPS reception means the receiver has a noisier signal to start with, and can't discipline its local oscillator as tightly. Even if you don't care about nanoseconds in the final application, you probably do care about _holdover performance_. And if your LO can be disciplined with nanosecond-accurate signals in the first place, it'll hold millisecond-accurate timing for days or weeks in a GPS outage. If the disciplining process is crap, the holdover will also be crap.