This turned out to be a nontrivial task. After hours of browsing and listening to advices (thanks to my friends who already gave some!), I see the following options:

1. Buy a laptop and use it as desktop.

Bad: it is not really silent; I do not know a place to buy a Linux laptop (meaning, with 100% Linux HW support, and without any other OS, for which I do not want to pay a cent, neither increase their sales statistics)
Good: it can be carried away if needed; it has the battery for power backup.

2. Buy a Mac Mini.

Good: It has just perfect form factor and hardware collection for my taste.
Bad: Despite some recommendations, I’ve not got enough interest to try it’s original OS. It seems that I’d have to replace the key native applications, such as web browser and media player, with opensource analogs anyway, and lose the (questioned anyway) “integrity” of the UI. I have not read anywhere that the HW is fully supported in Linux. And it looks like the not-so-ugly-as-everyone-else case costs quite noticeable money (if I take the idea that I’m not paying for the software).

3. Find exactly what I need ready-made.

There is a number of “thin client”-type Linux boxes on the market, but I’m looking for something bigger: I naturally need hard drive and a CD-ROM. And here the offer does not seem to exist at all. For example, a nice hit could be eBox, but it does not have the CD-ROM. The only 100% match I found so far is the carTFT series, mini-ITX PCs seemingly designed for car usage (shudder). They look like perfect match to my needs, except memory is limited to 1GB – but I probably put up with this.

Thanks in advance for any relevant ideas or personal experience.