Please help us help you and fill where relevant:
Your LimeSurvey version: 6.5.3+240415
Own server or LimeSurvey hosting: hosted by my university
Survey theme/template:
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Is it possible to make a question in LimeSurvey with the following layout:

• each row represents a single item of clothing (item 1, item 2, item 3, item 4, ...)
• each colum represents a different aspect of the clothing item (category of clothing, type of garment, type of fabric, ...)
• in each cell there should be a drop down menu which is different for each row. For each item, the participants select the category of clothing (the options for the drop down menu should be "adult clothing" and "children's clothing"), the type of garment ("T-shirt, top or blouse", "skirt", "pants", ...) and so on

This is what the full layout should look like (all the options for each dropdown menu are listed in the cells for each column and row):
 

Is it possible to do that in limesurvey? My university also only hosts the basic LimeSurvey version so I can't do any programming in the "script" window.

Hi,
yes this is possible - and shown many times in the forum.
  

Here I additionally disabled not applicable columns and showed a "dynamical" display of rows.

BUT:
If you can't use javascript

My university also only hosts the basic LimeSurvey version so I can't do any programming in the "script" window.

you unfortunately have to look for a different solution.

I could imagine splitting the question.
One group for each item.
If there is no more item do not display the following groups.
 
I have to admit: This is a very rough example - but the css snippet "no-bottom" was in this theme.

Joffm


 

My university also only hosts the basic LimeSurvey version so I can't do any programming in the "script" window.

If your university host the survey themselves (not via Limesurvey GmbH), there is no "basic" version of the community edition.

Maybe the have the XSS filter on, which prevents you from using Javascript.

Yeah, I messaged them if there was any way I could use javascript and they said no there is nothing they can do because they only provide the basic version of LimeSurvey. So not much I can do about this, I guess. 

But how does your university handle this?
Basic plan means: 1000 responses/month.
So they can run about 2-3 small surveys.

Anyway:
I showed a possible way to achieve your goal.
Which in my opinion is better regarding the respondent.
Each Dropdown requires two clicks, bootstrap buttons only one.

Joffm 

I honestly have no idea how my university handles this.

Yes, thank you for your solution!