The median price in East Cambridge (closest filter option on Hotpads) is around $3750. I used this search which removes subleases and rooms for rent, and requires photos/price so it should somewhat legit, but keep in mind real estate agents and potential landlords fuck with listings all the time to inflate the cost here or there. This is also the median price for all housing types - homes, apartment buildings, double/triple deckers, town houses, etc plus all bedroom/bath combos. If you're after a particular style, you can get a better answer by filtering. For example, studio apartments are more like $3000/month typically. 3 Beds are more like $4750/month. Etc.

YMMV if you have roommates, an SO, etc. Subleases can be cheaper in the short term too, and may give you some flexibility to hunt around more.

As for whether a post is legit or not, you'll need to:

• see if it's by a landlord or a real estate agent
• if it's the landlord, you can go to the Cambridge MA Property Database and look them up, or look up the address you're interested in. If they don't show up, you'll need to verify they're actually authorized by the landlord to lease the place to you. Double check if it's a relative, property manager, etc. You can also compare to the Middlesex County Registry of Deeds: https://www.masslandrecords.com/MiddlesexSouth/default.aspx?AspxAutoDetectCookieSupport=1 which is sometimes more up to date, especially if the property sold recently.
• if it's a real estate agent, Google their firm and see if they're legit or not. Do they have dozens or hundreds of positive Google/Yelp/whatever reviews? Or are they a fly by night or newer firm? You'll want to be extra cautious if you can't find much info on them.
• in either case, verify the address on Google Maps is legit. Drive, walk or bike by it if you can to confirm it still exists today and didn't burn down yesterday. Double check Airbnb listings to see if any photos look similar - a common scam is someone rents an Airbnb, then posts a video walk through of it and claims to be the landlord/agent/property manager/etc.
• if everything checks out, physically see the property in person for one final "yes, it's real, and yes, this person can show me the place". It wouldn't hurt to still confirm the person you're talking to is who they say they are via IDs/business cards/name on the door/etc.

It's hard to fake all of the above; very easy to fake a listing or video though. If you aren't able to physically see a property too, consider using a licensed broker - one with a lot of reviews and recommendations might help you look from afar with less chance of getting scammed if you look on your own. Double check their fee(s) and confirm ahead of time how it works. Gl.