I'm a British passport holder and need to find out an expired passport's number for a passport I held with a business visa in it. I no longer have the physical copy and speaking to the Passport Office and the Home Office has proved a fruitless exercise. As I spent a period of time in India in 2001, I need to obtain an Indian Police Clearance Check and my old passport number is necessary for this. Any advice please?
If your business visa was obtained while working for a UK company, or if you worked at a UK company after 2005, then the Human Resources department of that company would have taken a photocopy of the portrait page. You can contact the HR department and ask them to give you a copy of their copy, or simply to tell you the passport number. They are only obliged to keep records for 6 years (or 14 in some cases) so your luck may hold, or it may not.
Next. If you had pages of your passport certified by a solicitor or notary, they would also have taken a photocopy of the portrait page. Law firms have a varying retention framework unlike companies, and many like to preserve their archives for a long time.
If these fail, and as a last resort, you can file a Subject Access Request with the Home Office. There is a fee for this and you must clear hurdles imposed by the Data Protection Act. A Subject Access Request obliges the Home Office to give you their records about you along with what they know about you... if they want to. The SAR must be in writing, and as you have already observed, calling them is a waste of time.
If all of these do not bear fruit, then it would be reasonable to enter all '9's in the passport field and include an explanation.