Yes, such a system exists, for example the Honeywell WBS.* It was developed during the sixties and installed as an option on the Boeing 707-300 freight aircraft. Other aircraft that had a similar (optional) system include the L1011 and Boeing 747.
It is mostly found on freight aircraft because they have a less predictable load distribution (centre of gravity) than passenger aircraft.
The reason they are not common is that they add cost in the form of installation, weight and maintenance. While quite accurate when in good condition, the reliability of the weighing systems used to be poor, which probably contributed to the fact they never became wide spread.
Because crew are commonly operating on several aircraft types these days that have take-off weights of more than 100 metric tonne difference (e.g. A330 and A340, B747-400 and B747-8) there is an increase in take-off weight confusion related incidents. This, combined with the fact that systems are becoming more reliable, makes it likely that these weighing systems will be used more in the future.
* An advert for it in a 1987 Flight issue.
