This app was mentioned in 5 comments, with an average of 0.80 upvotes
I'd recommend this as well. You can get a cheap VPS server, setup SSH tunneling, then use an APP like this:
https://play.google.com/store/apps/details?id=org.sshtunnel&hl=en
And it'll do exactly what you want.
Root is not required. I don't know more than that, read the reviews:
You can also set up SSH tunnels, which can be a simpler option, depending on your client/server setup. They are similar to a VPN, but are much easier to set up IMO. Very simple server setup (most distributions come with a decent config out of the box), one simple command for generating key pairs. You can use them to wrap one specific service, or like a VPN, for entering the LAN remotely. For Android, app here, source here.
Have you tried SSH Tunnel?