How big is a number with 280 million zeroes?

To put a number like this in context, you need to think about entropy. Entropy is a logarithmic measure of the number of possible states, and the number you give is comparable to the number of states corresponding to the entropy in a grain of dust at room temperature. It is much smaller. Even a few $\mathrm{billion~atoms}$ in a solid will have a greater number of states, that is, a greater entropy than $280$ $\mathrm{million}$ times $\mathrm{ln}(10)$.

A more tangible combinatorial example is that this is the number of possible $100$ $\mathrm{megabyte}$ files. $100$ $\mathrm{megabytes}$ is a reasonably large amount of data, it is comparable to the size of the genome, so the number you give is roughly the number of possible genomes (without constraint for these genomes making sense, but also genomes can be longer than $100$ $\mathrm{megabytes}$, especially plant genomes, so it is probably roughly the right order). So this is more or less the number of different imaginable species of life on Earth, according to the current design of cells.

Another comparison is to text. Text has an entropy of about $1$ $\mathrm{bit~per~character}$, so to get $100$ $\mathrm{megabytes}$, you need $800$ $\mathrm{million~words}$, or $4$ $\mathrm{million~pages}$, give or take, or about $10,000$ $\mathrm{books}$, or a small library. This is the number of possible small libraries.

It is impossible to give this number an interpretation in terms of realized states, because there aren't this many realized states in the visible universe. So it can only represents number of possible states, or entropy.

The diameter of the Milky Way is 30 Kilo Parsecs or $9.25*10^{20}~\mathrm{meters}$
The diameter of the observable Universe is 29 Giga Parsecs or $8.8*10^{26}~\mathrm{meters}$
The age of universe is about $4.346*10^{17}~\mathrm{seconds}$
The upperbound of number of atoms in the universe is $10^{82}$
Combined wealth of all humans in the world is about $2.5*10^{14}~\mathrm{dollars}$
Total information produced by the human civilization is about 295 exabytes (as of 2011) or $3.4*10^{20}~\mathrm{bytes}$

Googol ($10^{100}$) is probably the maximum that you will encounter in any measurement in any applied science. In short, your $10^{280,000,000}$ is out of bounds in any practical science and exists only in the realm of theoretical math.

Not so for combinatorial stuff. If you look at numbers of possibilities, it is easily bigger than this.

As I said, in theory there are an infinite numbers big than these. But, even with the the practical applications of combinatorial stuff (such as breaking ciphers and keeping security in general) the number given is way bigger. Is there any practical problem where a number bigger than this is used?

I gave two examples: numbers of microstates (entropy in multiplicative terms) are often bigger than this--- any macroscopic entropy is bigger than this by a lot, and in evolution, the global search space for possible genomes is of this magnitude. I think the question was motivated by such a thing: the number is comparable to the number of possible different genomes, so it is usual to search through such a large number of possibilities in biology.

It's because of the mismatch between this number and physical numbers that you need stochastic algorithms to evolve new design, which can efficiently search such impossible space size.