I have a problem with exporting an Inkscape file to pdf. When open the pdf in the browser (Chrome) or in Adobe Reader, narrow strokes become wider than they should be (and wider than in Inkscape), at low zoom levels. Is there any way to solve this?

Trying foxit reader maybe?

Not much you can do about it. There are transparency related issues when saving as pdf-s but my guess this time it's the viewer's rendering issue.
Could be the result of different anti-aliasing.

My experience is that pdf readers render lines less than one pixel (eg .001") at one pixel (and antialiased) on a display until zoomed more than a pixel width, but printers and laser-cuttters will likely regard the line at its designed width... so if you printer can print .001", that is what you should get.

If they are purely vertical and/or horizontal lines, you can try aligning them to a whole pixel value. Extensions menu > Modify Path > PixelSnap will do it automatically, as much as possible. But this really helps in very few cases. If you can show us the image, we might be able to give specific suggestions (assumning anything can be done in the first place - but we can only know something can be done, if we see the image).