Ordering Integrated circuit fabrication
Do such service exist? Why not?


Yes, such services exist.

I'll let "why not?" stand anyway, because if you ask "Why can't I use them?":

By the phsyics of the processes involved, however, there's practically no chance that you'll be able to afford this :(

Silicon chips are produced on wafers. Wafers are large discs of highly pure, prepared silicon, that are then exposed to lithography and selective doping and metallization. The masks necessary for that are immensely expensive to manufacture. So having your own wafer will set you back at least a couple 100 k$, depending on the process you need. Sure, then you'll have hundreds of your chip, but that isn't all that great if you needed 3.

This is, however, how everyone in the market (who doesn't have their own fabs) produces ICs: you call your favorite fabs, get an offer for the process, wafer size and yield you want, and then transfer files and a lot of money. The most well-known example of such contracting fabs is TSMC.

Then, there's services that put a couple of your IC on to a "omnibus" wafer together with other customer's ICs. That of course restricts your process to one of the most popular ones, because you need a lot of other people to share your wafer in order to control the cost.

All in all, you don't need custom ASICs, unless you know what that means.


  Why can't I use them?


Because IC design still requires a lot of very specific, pretty hard, knowledge that, frankly, you don't seem to possess. Since making a mistake is really expensive here (see above), your 3D printing service is a  nice analogy:

Imagine the first 3D print of someone would be a really complicated part. How large is the likelihood that they produced the perfect 3D model file that can actually be successfully printed without someone manually going through that file, adding support where necessary, taking away unprintable parts etc? Practically zero. Now make your 3D print 10,000 times more expensive  and you see why this will no option for a beginner.


  I want to build SoC that I would use to build a server/super computer. I'm trying to build a computer like raspberry pi. But want it smaller and faster. If I get to design my own CPU. I could build something that much better, then existing systems, because server on chips would take less space and power. At least for my programs. 


Aside from my slight doubts that you alone or friends will get anywhere near the performance that CPUs have achieved after 50 years of optimization, this means that "old" processes don't make any sense, since you need high clock rates and thermally manageable losses. 

That means you need to use competitive structure sizes – if you want to be faster than current hardware, you can't use transistors that are slower. So something like 22nm or 14nm is your target structure size.

Sadly, a single mask for these cost millions of dollars. So this is totally out of your reach, both from a competence (how many expert on optimizing of memory accesses, pipelines, reordering, high-speed busses, speculative computation, on complexity of implementation, CPU design, ISA design, VLSI, … do you have on your team?) and a financial perspective.

If you want to implement your own CPU, start small, and program it and simulate in software. Then, put it on an FPGA. After that is proven to work, you can port it to ASIC technologies, and not earlier.