New washing line
A drying system that does not stand on the floor. I am more pleased than you will be.
This is one of those posts I am considerably more excited about than anyone reading it. I finally have somewhere to hang washing that is not a rack standing in the middle of the floor.
Why it was needed
We had been expecting to change the heating system in the spring, which would have freed up space. That fell through. The scheme needed half the households in the neighbourhood to sign up, most other neighbourhoods managed it, ours did not, and with it went the drying space I had been counting on.
The plan had been a dryer plus a small rack beside it. There is not room for both, so the dryer is off. And since the blue cabinet went in, the old line had come down.

So: something else. It is odd and it is not standard, and it works.

The batten side
Take a piece of wood that has been lying about in the garage doing nothing.
Cut to length, then drill three pairs of holes, both ends and the middle, for the fixings.

Prime it, screw it to the wall on plugs, fill the fixing holes, paint it, then screw eye hooks in along it at regular intervals.

The other side
The opposite wall has the heater on it, and its side is steel, so that end of the line runs off heavy-duty magnets stuck straight to the metal. No drilling at all.
I tried the same trick on the side of the blue cabinet for the longer run. It did not work: the span was longer there and the line sagged badly under wet washing. So that end got holes drilled through the cabinet side with eye hooks through them, washers and nuts on the back.
The magnets are fine for a short span and not for a long one. That is the one thing to take away.
Stringing it
Thread the line back and forth between the two sides, hook to hook, and tie it off at the end.
Done. And nothing standing on the floor.