"As I said, what would it take to send a manned mission technologywise to reach a NEA?"

Assuming you found a good space rock, it would be about same technology needed to build a space station at Earth/Moon L points. And if you keep the mission time down to say a couple weeks you wouldn't even any more sheilding than what's on ISS- as long as total time in space for astronauts was less than say 1 month and you could manage any risk from solar flares, then radiation levels would be tolerable.
This means you could use something like Mini Station 1:
http://www.mir-corp.com/news-press04.html.
If you stage from L-1 (leave for asteriod and then return to L-1) then you don't need to bring a heat sheild for Earth re-entry with you on trip to the space rock.
Sequence:
Launch the Mini Station 1 type spacecraft unmanned to L-1. This station is said to cost 100 million for LEO, so for any mods and allowing for extra cost, say the total cost would be 250 million.
Crew is launched via a Soyuz to LEO and docks to booster which lifts to L-1. [or perhaps using larger booster such as Proton goes directly to L-1].Costs about 100 million.
Add 100 million for separate supply launch which could have additional equipment needed to explore the asteriod (couple tons of stuff plus couple tons of booster fuel) this docks to Mini station.
Crew docks with Mini station at L-1 and then leaves Soyuz at L-1 using mini station to travels to asteriod and then returns back to L-1 and leaves Mini station at L-1 and uses Soyuz to return to Earth.
Total mission cost could be 1/2 to 1 billion.