There are many companies that do services for fibre from the cabinet to the home, but no services that do cabling from the exchange to the cabinet.
Is it possible to do that cabling personally or is there a possible way to hire BT to just install that part of cabling?
My neighbour hood is in close proximity to the exchange, so within about 400m.
Not exactly sure what you are asking here, your question is confusing, but I assume you are connected directly to the exchange rather than a cabinet (known as an EO line) and wish to have a VDSL (Superfast service). This is not technically possible as the equipment to provide VDSL services is not allowed to be installed in Exchanges for technical reasons. Occasionally Openreach have provided a cabinet in the Exchange grounds to provide VDSL but it is not universal. The cost of providing the cabinet runs into tens of thousands of pounds.
So basically you are asking about enabling your cabinet for FTTC, which again will cost tens of thousands of pounds. Have you asked your local BDUK scheme if there are any plans to fund your cabinet if Openreach don't intend to upgrade it?
Put you phone number in here and post a screenshot of the results. Delete your phone number from the screenshot before posting but leave exchange and cabinet information.
There are no companies that do fibre from the cabinet to the home. Not even Openreach.
Perhaps if the OP posted the requested information I asked for earlier we would have a better idea of the situation.