Let me try to (partially) answer the following question then:
Can I buy or build a quantum system, where I can perform certain quantum computations (maybe not universal) at home?
Tl;dr: The biggest caveat is the amount of control necessary to perform any controlled computation at all. You simply don't have it, except by studying a system with new technology for a long time.
I'm not an experimentalist, so my knowledge is limited, but I have some overview about ion traps and superconducting qubit proposals. The first is probably the furthest towards universal quantum computation and the second maybe the most promising (at least, there is a flurry of activity).
Before going into detail about the problem, let me mention that the D-Wave Two is probably the only commercial "quantum" computer out there. It doesn't claim to be universal, it maybe (or maybe not) does some "quantum computation" (not based on the circuit model, though), but it doesn't give any speedups. However, it gained a lot of attention as the first working computer - and still, its costs are far above anything you'll want to spend and it's also pretty big. This might already tell you, that it might be difficult.
Let's first discuss one of the major theoretical problems: Decoherence. You will never be able to fully isolate your qubits, so the state will decay - and quickly. In order to still do gates and computations, you'll need a) stable Qubits and b) Quantum Error Correction. If you really want to "play around with a gate" in the sense that you want to do (limited) computing and not just experimental control of single atoms/ions, etc. you'll need logical qubits, not just physical qubits. A logical qubit is a number of physical qubits where the state of one qubit is distributed between the physical qubits and error correction makes the time coherence time long enough to work with. In order to obtain a logical qubit, unless you have topological memories, you normally need to be able to implement the Clifford group (Paulis+Hadamard+CNOT) on aribtrary physical qubits - see the stabilizer codes for more details.
Then, you need not only gates, but fault tolerant gates, which is an additional requirement. Basically, if you tried to perform a computation with currently available physical qubits and gates, after a few steps (where "a few" means maybe a dozen or so) the outcome is completely random. That's bad for playing with circuits.
Let's have a look at ion traps first: One of the main points is the ion traps themselves. You'll need Paul traps that operate in vacuum. Chemists use them for spectroscopy very often, but you'll need more. You need them to be (and remain if not changed) in the ground state of the trap, i.e. you have to further reduce influences from the outside. This means a lot of calibration is necessary and you really need to "know" your system. It's experimentally very hard - what you want for yourself, however, is sort of a "black box", where you don't have to know of how precisely the lasers work, etc.
Now for superconducting qubits. As the name says, you'll need superconductors (Josephson junction), so you'll at least need to handle liquid nitrogen, probably even liquid helium - and that's bad for cheap "at home" experiments. You also need (as in ion traps) extremely precise materials. Have a look at this review (I couldn't find an open version of the ion-trap one): In figure 1, what you want is a system at step 6 or step 7. Maybe you'll also be happy with step 5, but anything else is not quantum computation. As you can see, as of 2013, no superconducting qubit system is even near that step in a lab. In particular, looking at table 1, there are fidelities listed for a few of the basic building blocks of superconducting qubits. Now, >0.98 seems great, but it's not enough for what you want to have. Using topological error correction codes, it might be nearly enough, for other types of error correction, you need much more.
For quantum computers, we're maybe in a state that the classical computer was mid 40s: The general idea was known, but we didn't have transistors, so playing around with it is extremely difficult.
What is possible then? If you just want to simulate specific quantum systems, you might be able to get simulators pretty soon. Maybe also some that don't require a whole lab to run, but I don't know. Simulators are limited quantum computers exploiting the fact that a quantum system can efficiently simulate itself, while a classical computer can't efficiently simulate most quantum systems. These are not based on circuit models or the like, they are specific systems, where you can simulate a very limited amount of other systems because they look somehow similar, so it doesn't really do what you want.