Why can't I declare a constant using var in C#?

The var keyword was intended to save you from writing long complex typenames, which cannot be constants.

It is very convenient to be able to write declarations like

var dict = new Dictionary<string, List<Definition>>();

It becomes necessary when using anonymous types.

For constants, this isn't an issue.
The longest built-in typename with constant literals is decimal; that's not a very long name.

It is possible to have arbitrarily long enum names which can be used as constants, but the C# compiler team apparently wasn't concerned for that.
For one thing, if you're making a constant enum value, you might as well put it in the enum.
Also, enum names shouldn't be too long. (Unlike complex generic types, which can and frequently should)

It is a compiler limitation, and the reason for that limitation is given by Eric Lippert here

Constants without var:

const int Value1 = 1;
const int Value2 = 2;

Constants with var (anonymous type property values cannot be changed after creation):

var constants = new { 
  Value1 = 1, 
  Value2 = 2,
};
//use as constants.Value1