You can't really find a place by yourself for that price in Cambridge (regardless of the dog). Median studio rent in Cambridge is $2,700 and 1-bedroom is $2,750 (https://www.zillow.com/rental-manager/market-trends/cambridge-ma/?bedrooms=1).

There's always a possibility you might be able to find something for $1,800-$2,000 if you're willing to accept a small/basement/a-bit-crappy kinda place.

If you're willing to live in Somerville (right next door) and willing to pay $2,000 you might have better luck.

It's always possible that you might find a place priced waaaaayyyyy under market, but it's just incredibly unlikely. If we could all get 1-bedrooms for $1,400-1,550, we wouldn't be paying $2,700.

Even in Medford, the average 1 bedroom is $2,300 with almost nothing below $1,900. Quincy, the average 1 bedroom is $2,150 with almost nothing below $1,600. Framingham, the average falls to $1,850 so you start getting a better chance out there.

Have you thought about roommates?