Let's get straight to the happy ending... I adore 57s, regardless of all the caveats, and all the learning, and all the hoops you may have to jump through. For me, the 57 does certain things in a way that is still unrivalled. That's remarkable for a speaker that was invented 70 years ago, but I have no doubt about their ability to compete with modern designs.
Practicalities
I do think they sound their best with room behind them (3 feet, ideally more), but some say you can mitigate that with EQ if you don't have the space (I've not tried it). They are BIG (sometimes described by detractors as panel heaters or radiators). They need fairly careful placement due to their relatively narrow dispersion (but not as narrow as some would say). You also need to take care over your listening position, for the same reason. I agree with
@Bencat that they need to be off the floor on a proper stand. Whether or not they need tilting possibly depends on how far away you are listening from and what height you're ears are at.
Myths
You will be told that they have no bass, you have to keep your head still and in 'the exact position' for listening, that they have no dynamics, they sound too small, they can't image. Pairs in less than optimal electrical condition may show some of those traits. Don't judge on just one listen to one random pair, you'll be missing out.
Buying
Whatever pair of 57s you buy, you are buying a restoration. That's just an unavoidable reality, given that panels and other parts all require repair / replacement after a number of years. To complicate matters, there are different replacement panels (and other components) available from different suppliers. Some are better sounding / more robust than others.
There are a lot of pairs available, at forums and on eBay. However, unless you are a 57s expert, you may struggle to accurately assess them. You can make sure they are 'working', you can check how quickly they 'come up' (how long it takes for them to play music after they are switched on from 'cold'), you can make sure there are no clicks or pops from the panels, you can check that each speaker is putting out the same volume as the other. What is more difficult though, especially if you haven't heard a number of different 57s, is knowing just how 'good' the pair in front of you actually are. 57s can still sound very good, even when they are 'a bit off'.
You could buy a set of very clean 57s in working order for, say, £600 to £800, and they might need new panels in 6 months time. On the other hand, they may give you several years of enjoyment without any attention. That's the risk. In any event, at some point in time they will need more money spent on them. That, for me, is why I reckon the smartest move is to pick up a set of 'non-runners', with the proviso that they are in good cosmetic condition. All the insides can be replaced, but the covers and the side cheeks are what they are (they can be improved a little, but not completely).
For my money (having learnt the hard way) I would take my time, and I would pay no more than £200 for a set of non-runners. Then, being in the UK, I would take them to 'A Quad Thing Audio' for a full rebuild. Their panels (and their new system they have for gluing them) are producing the best bass and treble that I have heard from 57s, and their all-round knowledge is second to none.
Sound
...is, the sound of no box. After a short time with 57s, you will know exactly what is wrong (to a greater or lesser extent) with all box speakers. That's not a dig at box speakers; all speakers are a compromise of some sort. 57s are super fast; there is no blur or overhang or muddle, and timing and transparency are superb. You will hear deep into the recording, but without fatigue or artificial brightness. They have a reputation for being at their best with acoustic instruments and vocals, and they are undoubtedly superb at reproducing those. If you're into heavy rock etc, then the lack of box 'thud' may throw you for a while. Mine go down to around 38hz in my room, that's very acceptable, to me. Others will want more bass... the difficulty is that I don't think there is a moving coil driver subwoofer that can keep pace with the 57's panels. Imaging can also take a bit of getting used to; the 57s radiate (front and rear) rather than 'point & squirt'.
Overall, the 57s sound like there's less 'hi-fi' in-between you and your music. I can't really give any item a higher compliment than that. If your room (and your partner) can accommodate them, then for around £2k you can have a fully rebuilt pair that won't be bettered, imho, for 5 times that amount.
So much has been written about them, and there's so much to say about them, I've barely scratched the surface in this post. Others will, hopefully, chip in.