Hello, Rhon_10N.
Without knowing exactly which Mac model and year that you have, I can only give generalities here. But yes, if your machine is not listed as being compatible with Big Sur, then it is not and you will not be able to install and run it. You may download the installer, but that is all. There is nothing else you can do. There is no workaround except to buy a new or newer Mac.
The macOS version you can upgrade to depends on the year your Mac was released.
Click the (Apple) menu at the top left of your screen and choose About This Mac.
macOS 10.15 Catalina is compatible with the Macs listed here:
https://support.apple.com/en-us/HT210222
macOS 11 Big Sur is compatible with the Macs listed here:
https://support.apple.com/en-us/HT211238
If you don't have enough room to download the installer, then I would say that your Mac is critically low on drive storage space. You should offload enough data to free up at least 20GB of space. Always having 10%-15% of total drive capacity free is ideal. If you cannot move enough data to the cloud, then consider buying an external drive for file storage.
Free up storage space on your Mac - Apple Support