There's that 'orrible "aitch word" again: headspace. Headspace on the LEE-Enfield rifle (and I have no idea why lazy people insist on calling them 'Enfields': Enfield made and designed a LOT of rifles over a period of 170 years) is one of the most misunderstood issues in the book. There are rifles with headspace 'way over book measurement which are safe with good ammo and shoot to beat the band. There are also rifles with perfect headspace that can't shoot worth beans.
Get a good barrel. If it's good at the muzzle, not oversized, nice sharp lands and grooves (especially just forward of the chamber) it can be MADE to shoot. If you have a good bore to start with, you're on your way. Likely you won't have bore gauges. I have bore gauges but they are finicky and slow to use accurately, so I carry an original Defence Industries (Canadian

eh! DI 19XX Z headstamp) live round with me. If I see a rifle I like, I pop the slug into the muzzle and see how far in it goes; with a minimum-spec barrel, the case should stand about 3/16" (3 to 4 mm) proud of the muzzle. If the casing contacts the muzzle, you're looking at toast. EVEN THEN, some of the 'toast' barrels can be MADE to shoot if you know what you're doing. I have a rifle here which was made in Australia in 1918 and managed to get through 2 World Wars and 2 rebuilds without being shot. When I got it, it was unfired. On test, it shot 14 inches at 100 yards. After some very careful diddling about and the expenditure of 1/4 sheet of 180-grit sandpaper and 2 dollars worth of Acra-Glas, it shoots just under half an inch if you can hold it and the Ammo is up to snuff. That's a 30-to-1 improvement, all the way from 'junk' to 'better than most snipers'.
LEARN all you can. Scoot on over to milsurps.com, take out a (free) membership, get into the Lee-Enfield Knowledge Library and start reading. Download and read some books, starting with:
RIFLE 1942 (Basic British Army manual for WW2)
THE LEE-ENFIELD RIFLE (1960) by Major EGB Reynolds (the ONLY history of the rifles ever done before all the paperwork was shredded)
SHOOT TO LIVE (the Johnson Method of Musketry Coaching, also the best introduction to Service Rifle shooting ever written: Canadian, eh.)
HATCHER`S NOTEBOOK by Major-General Julian S. Hatcher. Hatcher was the man who discovered the problem which led to a million Springfields being scrapped. He also founded the entire science of Forensic Firearms Identification. His book is the single finest compendium of firearms knowledge ever assembled between two covers and it is simply written and easy to read. It is also ABSOLUTELY authoritative.
THEN start downloading some of Captain Peter Laidler`s articles. Before retirement, Captain Laidler was the top Armourer in the British Army and he wrote reams and reams of essential knowledge down, likely more than any other single Armourer has ever done before. He is also on-line at times to help people with their impossible problems, as are THREE other full-time Armourers. I just doesn`t get any better than that.
One word: AVOID LIKE THE PLAGUE any rifle with ZF painted on the butt in letters about an inch and a half (4cm) high. It means that the rifle cannot be repaired at anything below Factory level...... and the Factory closed down 30 years ago!
Pick one you like which has a decent-looking bore that gauges well. If it doesn`t shoot, then get back to us here or over at milsurps and somebody will walk you through getting it up to snuff.
And don`t expect a 95-year-old rifle with iron sights which has been through 3 Wars and half a dozen Police Actions and Insurrections to shoot like a brand-new scoped $8000 tack-driver, especially not when using 75-year-old ammunition that Uncle Bob stored out in the chicken-coop. NEW factory ammo is for getting good reloading brass and that`s all; you will ONLY get top accuracy from your rifle with handloads. Serious. I haven`t bought a box of factory .303 ammo in the last 30 years for anything but test purposes.
Hope this helps.