Following TINAS’s example, if you don’t demand the latest kit then you could get close I reckon. A month or two back a mate and I were speccing his rigid 1×9 scandal build (all new kit). Along the lines of…
Scandal frame (£150), Exotic Carbon forks (£90), Superstar Wheelset (£200), Tyres (£50), XT cranks (£85), BB (£12), Cassette (£25), rear mech (£33), shifter (£30), chain (£11), brakeset (£180), KCNC stem (£25), seatpost (£40), Bars (£40)…
Add in a headset (£30), cables and outers(£15), seatclamp (£10), grips (£10) and a guide (£20) and you’re over £1000 – things can get expensive pretty rapidly. You might spare a bit of cash on the above by swapping things and finding bargains and still be around the target weight though. Having said that, the brakes and geometry would be the major differences between this and a 90s bike.