J, 17 bytes

#<.(-:'ab'#~-:@#)

This works correctly for giving falsey for the empty string. Error is falsey.

Old versions:

-:'ab'#~-:@#
2&#-:'ab'#~#   NB. thanks to miles

Proof and explanation

The main verb is a fork consisting of these three verbs:

# <. (-:'ab'#~-:@#)

This means, "The lesser of (<.) the length (#) and the result of the right tine ((-:'ab'#~-:@#))".

The right tine is a 4-train, consisting of:

(-:) ('ab') (#~) (-:@#)

Let k represent our input. Then, this is equivalent to:

k -: ('ab' #~ -:@#) k

-: is the match operator, so the leading -: tests for invariance under the monadic fork 'ab' #~ -:@#.

Since the left tine of the fork is a verb, it becomes a constant function. So, the fork is equivalent to:

'ab' #~ (-:@# k)

The right tine of the fork halves (-:) the length (#) of k. Observe #:

   1 # 'ab'
'ab'
   2 # 'ab'
'aabb'
   3 # 'ab'
'aaabbb'
   'ab' #~ 3
'aaabbb'

Now, this is k only on valid inputs, so we are done here. # errors for odd-length strings, which never satisfies the language, so there we are also done.

Combined with the lesser of the length and this, the empty string, which is not a part of our language, yields its length, 0, and we are done with it all.

J, 17 15 bytes

-2 bytes thanks to Jonah!

#<.2&#-:'ab'#~#

This works correctly for giving falsey for the empty string. Error is falsey.

Other versions:

<e.'ab'<@#"{~#\       NB. alternate 15 bytes, thanks to Jonah
#<.(-:'ab'#~-:@#)

NB. the following do not handle the empty string correctly
-:'ab'#~-:@#
2&#-:'ab'#~#          NB. thanks to miles

Proof and explanation

Outdated, but applicable to the old 17 byte version.

The main verb is a fork consisting of these three verbs:

# <. (-:'ab'#~-:@#)

This means, "The lesser of (<.) the length (#) and the result of the right tine ((-:'ab'#~-:@#))".

The right tine is a 4-train, consisting of:

(-:) ('ab') (#~) (-:@#)

Let k represent our input. Then, this is equivalent to:

k -: ('ab' #~ -:@#) k

-: is the match operator, so the leading -: tests for invariance under the monadic fork 'ab' #~ -:@#.

Since the left tine of the fork is a verb, it becomes a constant function. So, the fork is equivalent to:

'ab' #~ (-:@# k)

The right tine of the fork halves (-:) the length (#) of k. Observe #:

   1 # 'ab'
'ab'
   2 # 'ab'
'aabb'
   3 # 'ab'
'aaabbb'
   'ab' #~ 3
'aaabbb'

Now, this is k only on valid inputs, so we are done here. # errors for odd-length strings, which never satisfies the language, so there we are also done.

Combined with the lesser of the length and this, the empty string, which is not a part of our language, yields its length, 0, and we are done with it all.

J, 12 bytes

-:'ab'#~-:@#

Or, alternative version which doesn't error for falsey inputs, thanks to miles:

2&#-:'ab'#~#

Proof and explanation

Error is considered falsey. This is a 4-train, consisting of:

(-:) ('ab') (#~) (-:@#)

Let k represent our input. Then, this is equivalent to:

k -: ('ab' #~ -:@#) k

-: is the match operator, so the leading -: tests for invariance under the monadic fork 'ab' #~ -:@#.

Since the left tine of the fork is a verb, it becomes a constant function. So, the fork is equivalent to:

'ab' #~ (-:@# k)

The right tine of the fork halves (-:) the length (#) of k. Observe #:

   1 # 'ab'
'ab'
   2 # 'ab'
'aabb'
   3 # 'ab'
'aaabbb'
   'ab' #~ 3
'aaabbb'

Now, this is k only on valid inputs, so we are done here. # errors for odd-length strings, which never satisfies the language, so there we are also done.

J, 17 bytes

#<.(-:'ab'#~-:@#)

This works correctly for giving falsey for the empty string. Error is falsey.

Old versions:

-:'ab'#~-:@#
2&#-:'ab'#~#   NB. thanks to miles

Proof and explanation

The main verb is a fork consisting of these three verbs:

# <. (-:'ab'#~-:@#)

This means, "The lesser of (<.) the length (#) and the result of the right tine ((-:'ab'#~-:@#))".

The right tine is a 4-train, consisting of:

(-:) ('ab') (#~) (-:@#)

Let k represent our input. Then, this is equivalent to:

k -: ('ab' #~ -:@#) k

-: is the match operator, so the leading -: tests for invariance under the monadic fork 'ab' #~ -:@#.

Since the left tine of the fork is a verb, it becomes a constant function. So, the fork is equivalent to:

'ab' #~ (-:@# k)

The right tine of the fork halves (-:) the length (#) of k. Observe #:

   1 # 'ab'
'ab'
   2 # 'ab'
'aabb'
   3 # 'ab'
'aaabbb'
   'ab' #~ 3
'aaabbb'

Now, this is k only on valid inputs, so we are done here. # errors for odd-length strings, which never satisfies the language, so there we are also done.

Combined with the lesser of the length and this, the empty string, which is not a part of our language, yields its length, 0, and we are done with it all.

J, 12 bytes

-:'ab'#~-:@#

Proof and explanation

Error is considered falsey. This is a 4-train, consisting of:

(-:) ('ab') (#~) (-:@#)

Let k represent our input. Then, this is equivalent to:

k -: ('ab' #~ -:@#) k

-: is the match operator, so the leading -: tests for invariance under the monadic fork 'ab' #~ -:@#.

Since the left tine of the fork is a verb, it becomes a constant function. So, the fork is equivalent to:

'ab' #~ (-:@# k)

The right tine of the fork halves (-:) the length (#) of k. Observe #:

   1 # 'ab'
'ab'
   2 # 'ab'
'aabb'
   3 # 'ab'
'aaabbb'
   'ab' #~ 3
'aaabbb'

Now, this is k only on valid inputs, so we are done here. # errors for odd-length strings, which never satisfies the language, so there we are also done.

J, 12 bytes

-:'ab'#~-:@#

Or, alternative version which doesn't error for falsey inputs, thanks to miles:

2&#-:'ab'#~#

Proof and explanation

Error is considered falsey. This is a 4-train, consisting of:

(-:) ('ab') (#~) (-:@#)

Let k represent our input. Then, this is equivalent to:

k -: ('ab' #~ -:@#) k

-: is the match operator, so the leading -: tests for invariance under the monadic fork 'ab' #~ -:@#.

Since the left tine of the fork is a verb, it becomes a constant function. So, the fork is equivalent to:

'ab' #~ (-:@# k)

The right tine of the fork halves (-:) the length (#) of k. Observe #:

   1 # 'ab'
'ab'
   2 # 'ab'
'aabb'
   3 # 'ab'
'aaabbb'
   'ab' #~ 3
'aaabbb'

Now, this is k only on valid inputs, so we are done here. # errors for odd-length strings, which never satisfies the language, so there we are also done.