Short answer: Time Machine sizes a backup by the whole disk it is copying, not by how full that disk is.
A 1 TB internal drive with 200 GB used still reads as a 1 TB job, and it wants headroom on top for history.
The rule people expect is "the backup disk must be bigger than my files". Time Machine's rule is closer to "the backup disk should be comfortably bigger than the disk being backed up", because it is not storing one copy. It is storing a copy plus as much history as it can fit, and it would rather refuse now than run out in three weeks.
The practical guidance Apple has given for years is a backup disk at least twice the size of the drive you are backing up. Two to three times is the number most people land on.
tmutil destinationinfo
lists the disks Time Machine is set to use. To see the free space on both sides plainly:
df -h
And to see how much of your disk is actually in use, which is the number people expect Time Machine to be working from and it is not:
du -sh -x /System/Volumes/Data 2>/dev/null
rsync, needs roughly what your files take, with no multiple of the drive they sit on.On a NAS or Time Capsule, Time Machine stores the backup inside a sparse bundle disk image, and that image has a maximum size set when it was created. If the message appears on a network share, the limit is usually the image and not the volume: the share can have terabytes free while the bundle inside it is full.
Some NAS software exposes that limit as a per-user quota you can raise. Others do not, and the answer is to delete the bundle and let Time Machine make a fresh one, which throws away all your history.
At least twice the size of the drive you are backing up, and two to three times is the usual advice. That is about history, not your current files: a backup with no room for old versions is a mirror, and a mirror cannot get back the file you deleted last week. There is a size table in how much space Time Machine needs.
Not directly. It thins automatically as the disk fills, keeping hourly backups for a day, daily for a month and weekly for as long as it can. There is no setting for "keep three months and no more".
It means those folders are not in the backup, so they cannot come back. Excluding a re-downloadable game library is sensible. Excluding your Documents folder to make the number fit is how people discover the exclusion list on the worst possible day.
No, encryption does not change the size meaningfully. Tick the box.
Yes. Anything writing to a plain folder needs roughly what your data takes. Copy Shelter works that way, and will write to a cloud folder you already pay for, which Time Machine cannot use at all.
Running out of room repeatedly? Backing up a Mac to iCloud Drive covers using storage you are already paying for.