To be frank if I was quoted such, my first question would be "show me how you came by such a large sum".
I don't have any idea about this particular case, but if it were a complex and high-end custom chair made to measure for one person that sounds about right. Obviously if things are being batched the price can come down dramatically.
£5000/120 is about £40 an hour, but it doesn't work out like that in real life. Most people running a small furniture-making business are doing well if they can get three solid days in the workshop a week to actually put in those hours. On top of this there is design time, marketing, quoting, accounts, web design, photography, dealing with suppliers, sourcing hardware, ordering timber, sending back timber that is not what you asked for, delivering pieces, visiting clients to quote, days spent quoting for jobs you don't end up getting etc, etc etc. And then the inevitable mistakes which you have to put right in your own time and can't charge for. All of this has to be paid for, but is not being charged for directly.
Then there is a need to pay for a workshop and possibly business rates (which depending on where you live can be eye-watteringly expensive). And electricity for running machines which is not getting cheaper. And heating a largish workspace so it is at least dry enough for the timber to stay at a sensible moisture level and you can feel your fingers. And then there's the day to day running of the workshop. This week alone I spent £40 on replacement filter elements, £35 on shellac, £30 on re-grinding blades. And then things break and need replacing. It's constant, all the time. It's amazing how much things like abrasives cost. And there may be loans to pay off for machinery. And insuring all that machinery, and power tools and hand tools, and a store of valuable timber, and pieces worth £000's of pounds in the workshop and in transit.
And then timber, which is not getting cheaper, and has to be top quality if you are going to charge the sort of prices you need to in order to do the job at all, which means a lot of wastage, and possibly a trip to a timber merchant who is not just down the road, or an expensive delivery from same. And for a chair with curved components you may be sawing curves from the solid, out of boards 3" or 4" thick which is damn expensive. It's surprising how much timber a single curvy chair can use.
Assuming that 120 hours is a reasonable time to make the chair, which it may well be, I would say that taking all that into account you might end up with, maybe £20 an hour, perhaps, out of the £40 you are charging to the client, if you are doing very well indeed. That sounds fair enough to me for skilled craftsmanship. I wouldn't grudge it to, say a piano tuner or a web designer. Go to anyone else who is actually making a living (I mean really making a living) doing custom fine-furniture making and nothing else, and that is the sort of price you will pay. I wish it could be less. How I wish it could be less.
All of which goes to show why making fine furniture for a living is so damn hard to make work, and why hardly anyone manages to make a living from it. Try it.