I'm in a very small department, about 15 people, and we spend most of the day in meetings that aren't necessary, either all together, 1 on 1s, or 3-4 person groups. I usually only get about 30 minutes at my desk between 9:30 and 5:00 to get any actual work done but people will ask me for updates on projects as if I've had much more time than that. Even if it's someone I've been in meetings with all day, they'll ask if I've made any progress since this morning and then get mad when I say that I haven't been to my desk yet. What can I do about this?
First it's important that you make it clear that this is an issue. So raise the point to everyone and especially your manager that you have too much meetings and need to get work done.
Then, decline meetings. If there is a big meeting you feel only partly relevant, ask if you are fine not coming and simply read a report. Sometimes, it is hard as there is peer pressure about participating to meetings, but if you feel some meeting is unnecessary or comes at the wrong time, it's most of the time OK to say you won't participate and that it's not negotiable.
It's also important you ask or provide alternate ways to communicate that aren't synchronous and can get the less urgent communication happen without interrupting your work: this is what mail, ticketing, chat etc. are made for. This will allow to further free you from 1-1 that could have been a mail, and because writing requires more effort, it will also reduce communication noise.
Finally, you should realize some meetings are important. Sometimes, the company undergo change that require synchronization, and participating that can be more valuable than your work output. So it's good to accept that some days, you won't be able to do as much as you would like.
Make a weekly plan of all the meetings you have, try to see which ones are mandatory and which not and with that you can have a good idea of how much "extra time" you can get. With that planning, you can talk to your manager to explain the situation, show your calendar and see together if you can skip some meetings and re-do your calendar so you have more time to focus on your desk job.
With this, you show that there's a problem, why does that happen, how does it affect your work and how it can be solved.
If your manager doesn't care you should start thinking about if that work suits you, because it will look like they want you to do extra hours to get your work done.
You can try declining invitations to meetings you don't think you need to attend. Especially if you are in a senior position this may be a good solution. It should be mentioned though that, as a general rule, the more senior role you have, the more meetings you will probably need to attend.
If you are in a junior position you can try discussing it during your 1:1 (sic!) with your boss. Maybe they don't know about the situation?
But honestly, the fact that you are expected to sit in so many meetings and your boss still expects you to work shows something is probably wrong with the organisational culture. I've worked at organisations at which there was plenty of meetings. I've also worked at some that were actually well-functioning, result-oriented organisations. The distinction was always clear.
You can try observing how your colleagues deal with the problem. If they stay longer to deal with the workload, you have your answer.
If you have the ability further out in your schedule, try blocking off 8 hours of your time per week with a fake meeting. I personally block of my afternoons on Tuesday and Wednesday so people looking at my calendar think I am busy. You will have that time available if something urgent comes up and you need a meeting, but for the most part that time will be for you. I prefer larger chunks of time to be blocked off so I can gain momentum and finish projects. Sometimes only blocking an hour off is useless because by the time you start gaining some traction on something you have another meeting to go to.
Is it necessary for you to attend each meeting? I find that when people schedule meetings, they tend to include anybody that could potentially have an interest in whatever is discussed, not because they really need you to attend.
What I would do is explicitly decline the meetings that you think don't need your presence, and perhaps send a vague "no time" message. If they really need you to be there, they will let you know. If at that point they still insist on your presence, you might want to talk to your manager about how you should divide your time between meetings and actual work.