Although I don't know what exact version of Joomla 3 you're running you will need Joomla 3.10 to migrate to Joomla 4. Joomla 3.10 is a current version (until September 2023), so there shouldn't be any reason for your host to ban your site from using it.
The issue is probably with PHP 7.4 which is now EOL, and your web host rightly blocked it although it's perhaps unfortunate if your site runs less than 3.10, or only runs on PHP 7.4 or has extensions that only run on PHP 7.4 or earlier.
Joomla 3.10 works on PHP 8.0.x and is the migration 'pathway' to Joomla 4. You don't do any updates via FTP, you have to use the Update component within Joomla 3.10.
Can you use the FPA utility (red box, top of page) and post the report results so we can get a better idea of your hosting environment, and identify the Joomla version you're using?
If your host won't let you use PHP 7.4, one solution would be to set up a local web server on your own PC. Restore a backup of your site to it, and then do the update to Joomla 3.10 there (from whatever version you're running as you stated you don't really know what it is). You can also run whatever PHP version you need to such as 7.4.x (your local server is independent of your web host). You can copy your site, using the methods described here:
https://docs.joomla.org/Copying_a_Joomla_website (you need to export the database, and import it to a new db on your local server, and download all the Joomla system files). If you happen to use Akeeba Backup, then this method is the easiest, as described in the aforementioned documentation.
I'd recommend WAMPServer as you can install additional modules - various PHP versions amongst other things. So you can get your site running on PHP 7.4 or PHP 8.0.x
See this:
https://wampserver.aviatechno.net/ for the main Wampserver applications, and its accompanying add-on modules.
