It's actually quite possible to get an unmanaged VPS for under $2 a month if you shop around. I'm currently renting one for use as a personal email server for £0.002 per hour, which works out to around $1.85 a month for 1 core, 1GB ram, 20GB storage, and 2TB bandwidth.

https://lowendbox.com/ is a good resource for finding cheap VPS offerings to mess around with.