Q:

What is the maximum separation between two entangled qubits that has been achieved experimentally?

Considering two entangled flying qubits, as far as I know, there is no physical limit for separating them without losing quantum information.
See: Is there any theoretical limit to the distance at which particles can remain entangled?

Question

What is the actual record in such distance using photons?  I have found a previous record in 143 km (Physicists break quantum teleportation distance)

What about using solid-state based qubits?

Can this suppose a limitation when constructing a solid-state quantum computer?

A:

I believe the current record is held by the Jian-Wei Pan group in China, who are able to generate entanglement via a satellite. The journal article is here, while there's plenty of media coverage that is a bit more accessible, e.g. New Scientist. This claims a distance of 1203km.

A:

Photons travel fast, and there's often the option to transfer their entanglement to solid state. Of course, the advantage of transferring entanglement to a solid state qubit is that one is able to operate with it (one- and two-qubit gates, for example) with ease and efficiency, whereas it is very hard to effect two-qubit quantum gates on photons themselves, for more on that see the answer to How do you apply a CNOT on polarization qubits? So, let us divide the answer into optical-solid-state hybrid approaches, purely optical approaches and purely solid-state approaches:

The optical-solid-state hybrid approach results in records such as this one from 2012: Heralded entanglement between solid-state qubits separated by 3 meters. For the solid-state part they employed Nitrogen-Vacancy centers, which are diamond defects with remarkable quantum coherence, even at high temperature (although this particular experiment is performed at low temperature). In this case, the quantum fidelity of the final entangled state is well above the classical limit of 0.5 but at the same time well below 0.9, meaning it's enough to demonstrate quantum effects, but not great in a practical sense. Apparently, imperfect  photon  indistinguishability is the main limitation to fidelity in this experiment, followed by errors  in  the  microwave  pulses that are used to rotate the readout bases of the two solid-state qubits. As a more recent update on where things could be headed towards with the hybrid approach, there's this Demonstration  of  Entanglement  Purification  and  Swapping Protocol to Design Quantum Repeater in IBM Quantum Computer. As far as I read it, it's not a complete demonstration, since it does not actually implement the photon-solid transfer but rather "design a quantum circuit which could in principle equivalently perform the main operations of a quantum repeater". For a perspective on the whole field of combining quantum communications with quantum computing, see Nature Photonic's Towards a global quantum network(arXiv version).
The purely optical record, as reported in his answer by @DaftWullie, is claimed by the Jian-Wei Pan group in China, who report entanglement over 1203 km via a satellite (Satellite-to-Ground Entanglement-Based Quantum Key Distribution). Because of the nature of photons, this is more useful for purely quantum communication purposes rather than for actual quantum computing.
On the purely solid-state approach, I found this letter to Nature Nanotechnology of 2012, Electrical control of a solid-state flying qubit (arXiv version) Yamamoto and coworkers reported the transport and manipulation of qubits over distances of 6 microns within 40 ps, in an Aharonov-Bohm rings (based on the Aharonov–Bohm_effect), connected to two-channel wires that have a tunable tunnel coupling between channels. They claim to be the first "demonstrations of scalable ‘flying qubit’ architectures—systems in which it is possible to perform quantum operations on qubits while they are being coherently transferred—in solid-state systems". According to Yamamoto et al., "These architectures allow for control over qubit separation and for non-local entanglement, which makes them more amenable to integration and scaling than static qubit approaches." 

All that being said, probably the best practical answer to the question, at least for now, is currently working quantum computers: since it is claimed that 16-qubit IBM universal quantum computer can be fully entangled, it seems that the maximum distance of entanglement in solid-state devices will not be a practical limitation for quantum computing (even without employing flying qubits). I suspect that scaling and protecting that entanglement, however, will not be trivial.

