Btukfyl

It seems odd to carry around a box of dirt for a year. If he was going to make any use of it, he'd need a lot. Is there any mention of how much dirt Sam got?

It's not big. When Galadriel gives it to Sam, she says:

'For you little gardener and lover of trees,' she said to Sam, 'I have only a small gift.' She put into his hand a little box of plain grey wood, unadorned save for a single silver rune upon the lid. 'Here is set G for Galadriel,' she said; 'but also it may stand for garden in your tongue. In this box there is earth from my orchard, and such blessing as Galadriel has still to bestow is upon it.

Much later, back in the Shire, we see:

Then suddenly one day, for he had been too busy for weeks to give a thought to his adventures, he remembered the gift of Galadriel. He brought the box out and showed it to the other Travellers (for so they were now called by everyone), and asked their advice.
'I wondered when you would think of it,' said Frodo. 'Open it!'
Inside it was filled with a grey dust, soft and fine, in the middle of which was a seed, like a small nut with a silver shale. 'What can I do with this?' said Sam.

So, there's nothing like a precise dimension, but it's a "little box of plain grey wood" which contains "grey dust, soft and fine" and a "seed like a small nut." He uses the dust very carefully, only a grain at a time:

So Sam planted saplings in all the places where specially beautiful or beloved trees had been destroyed, and he put a grain of the precious dust in the soil at the root of each. He went up and down the Shire in this labour; but if he paid special attention to Hobbiton and Bywater no one blamed him. And at the end he found that he still had a little of the dust left; so he went to the Three-Farthing Stone, which is as near the centre of the Shire as no matter, and cast it in the air with his blessing.

There's many grains of dust in even a small box of "fine grey dust" and if each grain corresponds to a tree which is planted personally by Sam he'd have a huge amount of work to use up even a small portion of dust. This, again, points to a small box.