I assume that your father was self-employed, which means that he is/was the business. If he stops trading then the business ends, full stop. There is nothing to stop you starting a similar business, but it would be a different business.However, all successful businesses build up goodwill . This is the reputation and good name of the business, that attracts customers to them rather than to a competitor. Goodwill can often be very valuable - it is essentially only the goodwill that makes a business worth more than the physical assets that it owns.If your father is willing he could give or sell you his name and goodwill, which would give your business a head start. But remember that if you start doing a bad job your goodwill can be lost very quickly. This may also have capital gains tax implications for one or both of you, so if you are serious about this you need to speak to an accountant.As far as running the business is concerned, there is nothing to stop you starting a plumbing business even though you are not a plumber yourself. But you need to work out how you can attract plumbers to work for you rather than being self-employed themselves. Although a lot of people prefer the security of paid employment and might be good plumbers but bad businessmen so this might not be too difficult. You also need to ensure that your business makes enough money to pay your staff and to pay you what you need/want.
One way of doing it will be to use your father s business name, contact local self employed plumbers and farm any work out to them.You will, of course, have to ensure their work is up to the same standard as your father. The invoice to the customer will go from you as you will need to add a %age to the plumbers bill to cover your overheads.,This way, you don t employ anyone as they remain self employed and you have a group of people to call on when jobs come in. Possibly your father can give you some plumbers to contact.Also check out the legal and insurance implications.
A lot of plumbing firms are one man bands and as such close down when the owner retires. If it is a firm that employs several people you could act as manager. You need to ask your father for advice on this one