Looks like a nice rifle, and priced well too.
I'm inclined to say the serial number is "X1345" which would align well with a 1942 production BSA Shirley Rifle according to Stratton's book on the Enfield and Skennerton's notes on the enfield. Maybe /u/Caedus_Vao knows.
This is a Lee Enfield No.4 MkI. This rifle's development began after WWI in an effort to improve the No1 series of rifles, a series of experimental models were developed. The No1 MkV and MkVI rifles would be developed with a rear receiver mounted sight, functionally similar to that used on the P14 rifle. The MkVI was slightly modified and adopted in 1939 as the No4 MkI. This rifle had a stiffer action, barrel, bayonet mounting system, and the improved receiver mounted rear sight. Production of this rifle would start in 1941 at the new ROF Fazakerly and Maltby factories as well as at the private BSA Co. Shirley. Additionally, production would be set up in Canada at the Dominion Small Arms Factory and the Savage Arms Company in the US. Later, as demand increased, the No4 MkI* was put into production which made some design changes to the No4 rifle help speed up production.
This particular rifle was made at Maltby in 1943, mid production. By now the No.4 MkI* was in production in the North American arsenals. The British arsenals would never actually adopt the MkI* formally, but they would make some design changes to the standard No.4 MkI rifles to increase production. The major difference between the No.4 MkI and MkI* rifles was the means of removing the bolt. The small push button between the charging bridge and rear sight pivot screw was eliminated on the MkI* rifles. Additionally, the sight seen on this rifle was the standard used on the MkI* version of the rifle. My rifle however has some of these features but retains the MkI bolt release.
This rifle has a Mk 2 fixed two position adjustable sight, appears to have been made by ROF Maltby. Typically these rifles were made with a Mk 3 or Mk 4 adjustable rear sight in 43-44 but these two position sights were used as needed to supplement adjustable sights. They could be replaced at a later date with an adjustable unit in the field, but often this didn't happen. Forgotten Weapons has a fantastic video on the topic:
The rear safety on the No4 rifles was similar to that on the No.1 rifles in that it would block the striker from moving and lock the bolt in place. This specific safety however is a little different. It is one of the wartime expedient varieties of safety that doesn't include the threads needed to insert/retract the locking bolt to prevent the bolt from rotating with the safety engaged. These 2nd variation safeties were later deemed unsafe and replaced with the type 3 expedient versions that incorporated that bolt lock.
Some resources worth mentioning:
Jay Currah's Lee-Enfield Rifle Website
British Enfield Rifles, Lee-Enfield No. 4 and No. 5 Rifles, Vol. 2
This is a good overview page on the Enfield
This is a good page with examples of various models and some background
This is a great and cheap book on the No.4 and No.5 Rifles and where I got the info on your rifle from.
The Wikipedia Page on the Lee Enfield is actually pretty good and gives some high level overview of the design changes, variation, and history.