As a practical matter, that is probably true enough, as the files are not going to be accessible from the web even though they are stored on a “web server”.
But the real answer to your question is probably NO, as your files are still accessable to “root” and, depending on your permissions, potentially other users on the server.
Setting the appropriate permissions will protect you other users sharing your server, but I know of no way to hide your files from the system administrator of the system on which you are a user. I am not trying to be nit-picky with the distinction, but your question did say “only I have access”.
As for whether or not it is “decently secure”, you have to decide what constitutes “decent” and who you want the files secured from. I would never store anything on a shared server that I did not want the system administrator to see, but if you just want to keep your files as private as they can be on any shared system, you are probably “ok” as long as you set your permissions properly.
You could also encrypt your files so that, though the system adminisrator could access them, he/she would most likely be unable to break the encryption and read them without the resources of (insert name of governmental intelligence agency here)