I come from computer science and would like to define "construct" like this:

A real decimal number is constructible if you can write a program that prints out digits of the number forever. A decimal point must be printed at some time. The program must be of finite fixed size but can use more and more memory without limit as further digits are computed.

Given this definition, it is possible to describe a number that exists, but is impossible to construct.

This is related to the Halting Problem. Like in any version of the HP, we assign a number to each possible computer program in a certain language. The Halting Problem Theorem then states that there is no finite program that can decide whether another program will halt or get caught in an infinite loop.

In this version, we then asks for the real number between $0$ and $1$ where digit number $n$ after the decimal point is $1$ if program number $n$ halts and $0$ otherwise.

Any given program $n$ will either halt, or it will not. This means the number is well defined and exists.

However, writing an actual program to compute it would violate the HPT, so that is impossible.