Consider the following.....
There are 34 known radioactive nuclides in total- found in nature and/or produced artificially in the laboratory - with half-lives exceeding 1 million years. Of these, 5 are produced by cosmic ray bombardment of matter (cosmic rays are protons or other atomic nuclei that travel through the universe with speeds near that of light). So let's look at the remaining 29.
Of these 29 radioactive nuclides, 18 have half-lives that exceed 80 million years, the remainder have shorter half-lives. A half-life is the time it takes for half of the radioactive material to decay to another elemental isotope. Of these 18 whose half-lives exceed 80 million years, ALL of them are found in rocks and dust grains within the solar system - ALL of them. Of the remainder with shorter half-lives, NONE of them are found naturally occuring in the solar system (only produced in a laboratory) - NONE of them.
We can detect the presence of these radioactive isotopes down to relative concentrations corresponding to the passing of 20 half-lives or so. 20 × 80 million yrs = 1.6 billion years. So the solar system must be AT LEAST THIS OLD, or else we would detect the presence of the isotopes with shorter half-lives.
Not just the presence (or lack thereof) of radioactive isotopes, but ratios between isotopes connected by a decay route (e.g., 238U and 234U) also indicates a very old earth. It takes about 10 half-lives of the daughter to pass before the (asymptotic) equilibrium isotope ratio is reached (equal to the ratio of half-lives between parent and daughter). There are 13 intermediate radioactive isotopes between 238U and 206Pb (lead) which is stable, and all have their equilbrium ratios. This is true for every other decay path from long lived radioactive isotope to stable element.
It is as simple as this - put aside all the concordant evidence of absolute ages from rocks collected from around the solar system- the fundamental building blocks themselves (the elements) - their presence and their relative abundance ratios - are telling us how long our solar system has been around. And then the Sun itself tells us the same story, but that's another article.
A longer, but much more complete discussion of this (with data tables) can be found here.