What do "orange" and "spindle-shanked beaux" mean in this quote?

The number of unfamiliar, archaic words collocated here makes it a challenge even to a native speaker.

In order of appearance (all definitions taken from ODO except where otherwise indicated):

Spindle-shanked means ‘having long, thin legs’. It’s rather a contemptuous term to use, whereby a person’s shanks (legs) are compared in shape to a spindle.

Beaux is the plural of beau, meaning ‘a rich, fashionable young man; a dandy’.

Close is a verb here, meaning ‘to come close together in contact or union; to join, unite, combine, coalesce, meet in a common centre’ (OED), in this case referring to close romantic/sexual contact.

Orange wench is an old-fashioned term for a young girl who sells oranges (and other fruits, presumably) in the streets.

In other words, the whole thing means, roughly translated into 21st-century English:

Our current race with its lanky, dolled up pretty-boys [who] would rather fiddle around with the greengrocer’s girl behind the local drive-in than get in on a backalley fight [for money] at a boxing match.

(See StoneyB’s answer for more details on the context of boxing matches)

The context of this quotation is fisticuffs, and "Tottenham Court" refers to the school where James Figg taught boxing, fencing and quarterstaff.

An 'orange wench' was a woman who sold oranges and other refreshments at a playhouse, and to 'close' with her means to engage her in conversation, probably with dishonorable intentions.

A 'spindle-shanked beau' is a fashionable gentleman with thin calves. Muscular calves were admired in the 18th century, a period when men's clothing was designed to display the lower leg.