What will happen if satellite navigation fails?
It only takes 3 satellites for a GPS fix (4 if you want altitude). There are 31 operational satellites currently in orbit and at any given time you can most likely receive a signal from half a dozen of them at least. That being said the chances of them failing completely is almost 0%. But of course what if they did, well...

There is always the Russian version of GPS known as GLONASS so there is effectually an entire redundant global navigation system in space. While not all aircraft are equipped with GLONASS receivers it is of course becoming more common. Even the little tiny receiver I use for GA flights has it now. So there is that redundancy.

In controlled airspace you have ATC there to help you. They can effectively watch you on radar and tell you what to do which is basically what flying IFR is. For this you will need an operational altitude encoding transmitter so that ATC knows your altitude along with position. If you are flying VFR and require assistance you could call for Flight Following here in the US, load permitting. They can offer limited guidance but if you tell them you are lost they will generally help.
If there is no ATC you can hone in on VOR stations, or some of the older radio direction finders like ADF systems assuming your aircraft is equipped with such equipment. Some modern handheld radios can also hone in on VOR stations

If all that fails you have your good old magnetic compass and hopefully some way of keeping time. With close track of your airspeed, a clock, and a compass you can navigate pretty accurately even in the blind (although I don't recommend that).

No compass, no problem, you always have your ground reference points to get where you need to be. Basic VFR navigation which every pilot know how to do is learned this way. This will require you have a VFR map of the area which most GA planes will carry. Note the roads, power lines and other large things marked on the map that are easily identified from the sky.

Outdated/less popular systems:
Inertial navigation systems are still used in plenty of places however they are not as accurate as other systems available in modern aircraft but they work by knowing how fast you have gone and in what direction you have accelerated to that speed. Thus position can be calculated. INS systems suffer from integration drift which can add up over time and make the margin of error pretty big.
There was once a time in history when you could get a navigators rating on your license. These people were trained in many times of navigation including celestial navigation. Older planes were even outfitted with a sextant for that very reason.

LORAN C was officially shut down here in the US/Canada in 2010 but there are still active chains elsewhere in the world. Some planes out there still have the units in them (one of the Cherokees I fly still has it sitting there). So depending on where you are you may be able to use it.
What Time Is It:
The US GPS network is synced up via the USNO master clock and while it is unlikely that all the satellites will fail simultaneously or fall out of the sky, or be shut down. The master clock does provide a central point of failure for the system. While I am not up to date on the full function of the system I would imagine that a downed clock could introduce a margin of error into the system. Worse so an incorrect value fed from the master system may cause an erroneous reading.