Is it safe to store the encrypted private key of a RSA pair on the same machine as the encrypted data?

Question

I was not sure how to google this. I want to set up a storage server where I encrypt everything that is sent to it, from anywhere. The files sent to it would automatically get encrypted by the public key. But in order to never lose the private key, and have easy access to it I would like to also store the encrypted private key on there. I would probably use something like AES256 to password encrypt the key. But is that safe? An alternative would be to keep the encrypted private key on a cloud service like dropbox.

Also, does this make sense to do? Can I have any suggestions to applications which might already do something like this.

The idea is that I can be on any computer, just send the file to my server (with some sort of authentication preferably). And it will encrypt it with the public key stored on the server. That way, even physically no one can grab that data except for me who has the password for the private key.

I am open to all suggestions and opinions.