You need to specify that the variables are vectors of a given dimension:

Reduce[{a . a == 2*2,
  Element[a, Vectors[2]]},
 a, Reals]

Your particular problems seems hard to solve symbolically, probably because VectorAngle involves trigonometric functions:

Reduce[{a . a == b . b == c . c == 2^2,
  VectorAngle[a, b] == \[Pi]/3,
  VectorAngle[a, c] == \[Pi]/3,
  VectorAngle[b, c] == \[Pi]/2,
  Element[a | b | c, Vectors[3]]},
 {a, b, c}, Reals]

I get a quick answer by manually simplifying VectorAngle and by setting the direction of the first vector and the plane of the second:

Reduce[{a . a == b . b == c . c == 2^2,
  a . {1, 0, 0}/(2*1) == 1,
  b . {0, 0, 1}/(2*1) == 0,
  a . b/(2*2) == Cos[\[Pi]/3],
  a . c/(2*2) == Cos[\[Pi]/3],
  b . c/(2*2) == Cos[\[Pi]/2],
  Element[a | b | c, Vectors[3]]},
 {a, b, c}, Reals]
% // ToRules
Norm[3 a/2 + b/2 + 3 c/2] /.
  Table[r -> Table[Indexed[r, n], {n, 3}],
   {r, {a, b, c}}] /. {%}