The system you want is ZF+ dependent choice + "All sets of reals are Lebesgue measurable". This is the comfortable system for physics.
The main theorems which are required for the development of analysis and physical mathematics only require dependent choice, not full continuum choice. The new axiom that all sets of reals are Lebesgue measurable then drastically simplifies measure theory and prevents paradoxes in probability that make intuitive arguments fail. This allows you to avoid the intuition traps that have forced rigorous mathematics and matheamtical physics to diverge.
For an example of such an intuition trap, consider R as a vector space over Q. Can you find an algebraic basis for this vector space?
The answer is obviously no, when you can speak naturally about probability, for the following reason: pick two independent Gaussian random numbers x and y, with unit variance. They have a certain probability distribution P(n), defined on positive integers n, of being made up of a certain number n basis elements.
Now combine them into another Gaussian random number of unit variance by adding them up with rational coefficients, for example z=(3x+4y)/5. Then z has a decomposition of m+n basis elements (where n is the number of basis elements of x and m is the number of basis elements of y). But z is again a Gaussian of unit variance! So the number of basis elements of z must have the same probability distribution as that of x and of y.
But you can easily prove that it is impossible for the sum of two independent positive numbers distributed with distribution P to have the same distribution P. The reason is simply that it is a probability distribution on positive integers, so if there is a nonzero probability at 1, the sum distribution is only nonzero at 2. In general, if it is first nonzero at N, then the sum distribution is first nonzero at 2N, and it cannot be the same distribution.
But the answer in standard ZFC is that yes, you can find an algebraic basis! It just simply happens that the sets involved in the argument are not measurable, so you are simply not allowed to speak about decomposing a randomly chosen Gaussian number into a basis, this concept is meaningless. This means that randomly chosen real numbers in ZFC are second-class citizens, you are not allowed to perform the same operations on such numbers as you would on numbers which are specified by set operations, rather than random picking.
Similar probabilistic disproofs, using randomly picked numbers, can be given to all the statements which are the counterintuitive consequences of choice. The Banach Tarsky theorem is an obvious example: if you pick a random number, the probability that it lands in two spheres is greater than the probability than it lands in one, so no transformation can map the two spheres onto one (except if you remove the ability to talk about randomly picked numbers).
Each of these intuition conflicts amounts to the intuition that it is possible to choose a number uniformly randomly from [0,1], for example by flipping coins for the binary digits. This is the key in Solovay's construction. When this is possible (it is always convergent), and when this generated number can be assigned membership to arbitrary sets, then it follows that every set is Lebesgue measurable. Conversely, if every set is Lebesgue measurable, it makes sense to speak about random uniform picks from [0,1] without contradiction, because every question you ask about this number has a well-defined probability of being true.
So in this kind of universe, you can talk coherently about random real numbers, about random Gaussian picks, and so on, without fear of hitting a contradiction. This is required for many constructions in statistical and quantum physics.
For example, this allows you to define path integrals and statistical limits, intuitively. To define the free scalar field, simply choose a Gaussian random real with appropriate variance, and do a Fourier transform. To define a stochastic differential equation, pick a noise, and solve the equation, and show that the limit exists. These measure theory arguments become super-annoying when there are non-measurable sets lurking about, because you need to prove that the non-measurable sets never appear when you are transforming your noise to your solution.
Using such a system does not discard any mathematics, you can still talk about classical choice constructions, because every model of ZF has an inner model (Godel's L) which obeys the axiom of choice. So within your Solovay measurable model, you still have a normal usual choice model of set theory. The theorems about Banach Tarsky then you simply reinterpret as theorems about the L submodel. The Banach Tarsky decomposition is then simply a decomposition of all the L-points of a sphere so that they are disjointly mapped to all the L-points of two spheres, something which is neither counterintuitive or surprising, because the L subuniverse is a small measure zero dust within the full universe.
The issue of which model best approximates something like "mathematical reality" is not one which has a logical positivist answer, but if you ask my own opinion, the measurable universe is always better thought of as reality. The constructions of objects which lead to non-measurable sets of reals are always in conflict with intuition, and are incompatible with a simple procedure which denies their existence. But these ridiculous constructions are self-consistent within an inner model, so that you do not lose the arguments made using them, you simply reinterpret those arguments as applying to an impoverished sub-universe within your true universe.
This has been advocated many times within mathematics, but mathematicians are conservative and stupid, and take forever to get with the ball. This should be changing now, but it shouldn't have taken 40 years.
As for the inaccessible cardinals required in the argument, this is a ridiculous propaganda made against these models. When you are proving relative consistency, you often need to go to models where you can explicitly describe a model of the previous axioms as a set. Within ZFC, the strongly inaccessible cardinal is simply the first normal set which can be used as a model for ZFC, and this is what it is used for by Solovay. The essential argument for why measurability is consistent, the fact that you can consistently pick random real numbers, is not itself at all sensitive to large cardinals, it's just that translating this construction into a full model of ZF you need to start in a universe where the ZF operations can be applied to a set which already makes a model for ZFC.