There is no law that states you may not have an aired & loaded marker beside your bed, neither is there a law that you cannot have an axe beside your door, and the same can probably be said about licenced firearms that have to be securely stored - other than the terms of the appropriate licence and the holder not being in control if they are asleep etc.
Police who are attending a property when there is an axe or baseball bat beside the door may have some thoughts, they will consider them against other factors - e.g. they will be a bit suspicious of the intent if they are breaking the door down of a drug dealer.
However if their are children with access to the property then the consideration of a paintball gun being an air weapon makes the legislation applicable.
Security Act 2010
http://www.legislation.gov.uk/ukpga/2010/17/section/46
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/117804/air-weapons-safety-leaflet.pdf
You can argue the finer points of paintball guns being air weapons or not. The UKPSF and the Home Office generally classify paintball guns under various aspects of legislation and they fall in or out on a number of criteria.
Air weapon and velocity do not necessarily apply because of a seperate case that treated them as gallery guns and took the precident of 300fps instead of translating the pressure of 6 or 12 ft/lbs
There are two key points in the legislation and guidance regarding storage - a key case in the lead up of the legislation was the death of a child that found their parents airgun, and the recommendation against storage in outhouses, sheds etc - these make it easier for the criminal element to break into the secure storage and possibly also use other tools in the shed.