You need to repair your polygon layer as it conatins many errors. See below how to do that. I did a "quick and dirty" repair, download the result here: https://drive.switch.ch/index.php/s/4MinvisRoHfmSxc - be aware: many of the tiny polygons were deleted, so make sure this is really what you need. In any case, with this polygon, you can clip a raster:
See the screenshot: I used a raster DEM to clip it with the result from above - the result are the grey areas, representing the clipped raster. The red areas are those polygons that were deleted in my repaired polygon above. If you have time, you could try to fix this red areas as well (see below how):
Clip raster by mask layer, I get the following error:
IllegalArgumentException: Invalid number of points in LinearRing found
3 - must be 0 or >= 4 ERROR 1: Cutline polygon is invalid.
So your layer contains invalid polygons (duplicate node, self contact), that's why it does not work. You have a very complex polygon layer, containing holes, slivers and a few just simple, but very tiny polygons etc. So there is a lot of work to clean it up.
To repair the polygons yourself, depending on what you want to achieve, consider using just a few larger polygons and get rid of the other ones.
Getting a valid polygon layer, you can do the following things:
Convert the multipart to single parts polygon layer, then create a new attribute calculating the area and delete all those polygons smaller than a certain threshold.
Buffer your polygons by a (not too large) amount - due to your data, I guess 1000 meters would be a good start point. After that buffer again with -1000 meters to get back to the original shape, but with some problematic areas eliminated. Be aware: your layer is in CRS
EPSG:4326, with distances in degrees. First reproject to a projected CRS before buffering.
Run Check Geometries plugin to repair the remaining polygons.