Is there an English word for "work which is repetitive (and often boring), but which must be done"?
There are several possibilities, depending on the occasion and nuance. My favorite basic one for a thing that is "a) repetitive, b) relatively boring, and c) has to be done" is chore.
Here is its definition in Macmillan Dictionary:
an ordinary job that must be done regularly
You can go play after you’ve done your chores.
The word is generally used for household chores, but it does get used in a variety of contexts – like in this news story about a retiring judge:
At first, the job was a chore, but as he saw how his work could help people caught in terrible situations, it became a mission.
The word chore is also ideal for describing "repetitive" tasks. Chores are by nature things that need to be done on a regular basis, whether it's referring to chores on the farm, or chores around the house.
To emphasise that the work is boring and without reward, consider one of:
Grunt work: work that is menial and thankless. During the summer, I earned money doing grunt work. I did all of the grunt work on the project, but my boss got all of the credit. The Free Dictionary
Grind: to do tedious, laborious, and sometimes menial work. The Free Dictionary
The latter has some nice idioms: back to the grind (going back to work after a break or vacation), daily grind (a tedious daily routine/job), grind away [at a job] (e.g. "he had to grind away for years at redecorating the old mansion"). You sometimes also see grind work and on your grind.
If you want to emphasise the ongoing physical difficulty of the work, and the exhaustion it can cause, toil (which can be used both as noun and verb) would be a good choice. The fact the work is mundane is often implied, but it isn't so explicit.
Toil: hard and continuous work; exhausting labor or effort. Dictionary.com
To emphasise that the work is continuous and unrelenting, consider working like a dog or (a common metaphor) being on the treadmill (which particularly hints at how one feels little sense of progress or having "got anywhere").
Work like a dog (UK also work like a Trojan): to work very hard. You can work like a dog and still not make ends meet. Cambridge Online
Treadmill: a monotonous routine or set of tasks. The Free Dictionary