How can it be that a machine that complicated can not know it's own weight and how much runway it needs to get to x speed?
The first aircraft which I encountered in service with a capability to weigh itself (and assess its own centre of gravity) before takeoff was the Vickers Vanguard Merchantman, which first flew in the late 1960s.
Why do modern aircraft not weigh themselves? adverse bump had it right in post two - it's about money.
Standard passenger and bag weights, used to calculate the aircraft's takeoff weight, are 'agreed' between operators and regulators, and then set out in regulation.
People and their bags very often weigh considerably more than these values, but little is done to assess this and correct the standard figures, because there would be negative implications for airline finances.
So, why is this not very risky? The two times when an overweight aircraft becomes a big problem are in the event of engine failure after takeoff with loss of thrust, and on a performance-limited landing. Engine failures are rare nowadays, and failures in the tricky regime between V1 and V2 are very rare indeed. Landing distances have considerable fudge factor, so overweight aircraft don't often overrun either. (Cynics would point out that by the time investigators get to the site of an overrun, the passengers and their bags have often left the scene, so assessing the landing weight is difficult).
There is nothing to stop modern aircraft being 'told' (by way of an onboard database) about runway lengths. Yes, the aircraft could do all of its own performance calculations. There would be no problem devising hardware and software to check weight, assess performance, and ensure the 'safety' of the proposed takeoff.
Even more importantly, aircraft could be fitted with takeoff performance monitoring systems, which would take all of this into account and ensure that the scheduled performance is met. The relatively high frequency of takeoff accidents and near-accidents (including the recent spate of Airbus problems) would count in favour of this proposal. However, like its friend the on-board weighing system, such a device would alert when the aircraft is overweight, and the airlines wouldn't like it.
Barry Sweedler of the NTSB famously said
'We regulate by counting tombstones', and while this technology is very feasible, there are not yet enough grieving families in the world to force its introduction.
Sweedler's words, moreover, were spoken at a time when aviation was statistically less safe in general, and nowadays accidents are sufficiently rare that I'm not sure his words carry the weight they once did.