If a solicitor has written a will, they will normally be able to store the original; this is generally part of their will writing package, although it is worth checking if storage involves an additional fee. DIY wills (eg the one provided by Rocket Lawyer) can also be stored by a solicitor, but this will normally incur a fee.
Will writing services
There are various companies offering will creation services (eg Co-op and Trust Inheritance) which additionally provide storage facilities. But unlike solicitors, will writing services are not all regulated, so it is important to keep a backup copy just in case.
In England and Wales, wills can be stored with the Probate Service for a one-off fee of £20. This is a formal process that requires a form to be submitted at a regional probate registry; alternatively, it can be sent by post using a safe custody envelope pack. Any changes to a will also need to be submitted to the Probate Service via a Codicil. See form PA7 for further information.
Some banks offer will writing and storage services. However, wills stored in a bank should be accessible to executors (ie safety deposit boxes are generally not suitable - see below). Banks in Scotland may charge an annual fee for this.
Keep it yourself
This is the cheapest and easiest solution - but it's also the riskiest. If storing a will at home, make sure the location is secure and known to executors.