Yes, doing it as the
root user would allow you to change the permissions on (and owner of) these files and directories.
$ cd /backup/dir/somewhere
$ sudo chown myname:mygroup myfile
$ sudo chmod u+rw myfile
mygroup is your username and default group (check you other files in your home directory or use
id -n -u (for username) and
id -n -g (for group)), and
myfile is a single file.
To do this on a whole directory, including its contents:
$ sudo chown -R myname:mygroup mydir
$ sudo chmod -R u+rw mydir
However, a directory needs to be "executable" for you to be able to
cd into it. The "write" permission on a directory allows you to add or remove files to/from it, once you've entered it. With the "read" permission on a directory, you're allowed to list its contents, once you've entered it. So only having "read" and/or "write" permissions on a directory makes no sense.
Just don't do
$ sudo chmod -R u+rwx mydir # don't
as that would make every file in there executable.