A proper EEG for use in sleep studies is available on ebay (last time I looked, maybe six months to a year ago) for $1000 for a USB model or $1500 for one with a battery (might require a prescription). There are also ongoing costs, since the standard usage is to stick the electrodes to a particular location and I'm not sure how accurate it would be if you didn't do that (it is possible that some of the long term EEG monitoring caps used for epilepsy might work with less setup time, but I don't know if it would be directly comparable to the usual method). IIRC, it takes about a half hour to get set up for a sleep study. Additionally, scoring the sleep study is not a fully automated process and without the same training that sleep doctors receive the results would not be comparable. Getting a reliable and useful result might be quite difficult and you would need to compare with regular sleep studies to see how accurate a particular method is.
There seem to be an endless stream of companies trying to make easier to use EEGs, either for clinical or home use, and it seems like almost all of them go bankrupt in a few years. I suspect most of them are also not very accurate. A couple of years ago I saw an interesting one that tried to automate most of the scoring but would send the more difficult cases for human scoring. I'd guess that one at least has a better chance of reasonable accuracy than most. I think it was dreem (dreem.com), although they only mention automated scoring at this point. Possibly they managed to train their automated system sufficiently or maybe it was a different company. In any case, I would be suspicious of claims that they are as accurate as a sleep study, although it is certainly possible that someone has come up with an actually accurate method. You would still be spending hundreds of dollars for a device that sends all the data it collects into the cloud and in most cases will be useless if the company goes out of business or decides to stop supporting older products. They usually aren't clear about what, if anything, will still work if the company goes out of business.
Of course, less than ideal results can still be interesting. Once a few years ago I managed to set up my laptop's microphone to record might sleep at just the right level and it was very interesting. Unfortunately, I didn't write down how I changed the settings to get the right volume and could reproduce it when I tried a couple of times, although I imaging there are online guides for this type of thing and I didn't go looking for them. For anyone who sleeps alone and does not use CPAP I'd recommend giving it a try. With builtin laptop speakers you can only hear when sleeping on one side (I never sleep on my back, so not sure if that would be covered), but I bet it wouldn't be too hard to set up two or three microphones to get a good recording in any position. It isn't trivial to review, but looking at the recording in audacity it is possible to see the regular parts and just listen to the less regular parts. You can't even fully tell asleep or not from that, but you can notice when your breathing changes and in my case it was very interesting to see how that often happened due to quiet noises (e.g. a dog barking in the distance, car going by quietly). You can also tell how often you move at night (about every 45 minutes in my case, at least that night).
I've also used a basic pulse oximiter (Contec CMS-50I, was about $90 new on ebay) with the Sleepyhead software. It has pulse, SpO2, and perfusion index. At quick count it looks like I've used it 24 nights over the past couple years and I'm not sure the device will necessarily last much longer, hopefully it will (the display has an always on line and there are sometimes issues with the sensor that might be due to an issue in the cord, although it is sometimes due to not being cleaned sufficiently or to my hand moving in the sensor; the sensor can be replaced for something like $20 and it came with a second "one time use" sensor that I haven't tried but I suspect might be possible to use more than once). The pulse reading seems to accurately show when I first get to sleep, but doesn't help determine when I am sleeping for the rest of the night. SpO2 is the main thing I use it for, but doesn't help determine anything about sleep. Perfusion index mostly seems to indicate how warm my hand is and so can sometimes indicate when I change position.