Only when the caster is below 5th level

The errata for the PHB has clarified the restriction further from the original printing:

To be eligible for Twinned Spell, a spell must be incapable of targeting more than one creature at the spell’s current level.

By default, eldritch blast does not have a range of self and is capable of targeting only one creature. It is thus eligible to be twinned.

However, the spell becomes capable of targeting more creatures once the caster reaches level 5:

The spell creates more than one beam when you reach higher levels: two beams at 5th level, three beams at 11th level, and four beams at 17th level. You can direct the beams at the same target or at different ones. Make a separate attack roll for each beam.

Thus, once the caster reaches 5th level and above, they can no longer twin eldritch blast.

Note that eldritch blast (and all other similar cantrips) scale with character level not class level as confirmed (unofficially) by Jeremy Crawford on Twitter. (See Do Cantrips use your character level or class level? for more discussion about that)

You cannot twin any of these spells

Eyebite has a range of self so it is explicitly disallowed when twinning as per the Twinned Spell description:

When you Cast a Spell that Targets only one creature and doesn’t have a range of self...

Bigby's Hand and Telekinesis do not target a creature, but rather each create an effect, which can then target a creature. This would seem to disallow either of them from being twinned.

In addition, they both could affect multiple creatures throughout the length of the spell, which should disqualify them as well. As per the Twinned Spell description,

To be eligible, a spell must be incapable of targeting more than one creature at the spell’s current level.

This does not say "incapable of affecting more than one creature at a time." Additionally, the extra rules that would be required in order to dictate how they work on future turns would likely be more than most tables might be comfortable with.