Whether or not the country could have afforded it is perhaps secondary to whether or not the South would have allowed it. Southern politicians had long opposed both most kinds of infrastructure spending and growth of the federal debt. They had also, in the twenty years before their attempted secession, violently attacked anyone and anything that dared to question the moral goodness of slavery. Southern politicians had, at various times and places, instituted censorship programs over the mail to intercept abolitionist literature coming into the South, instituted a gag rule in Congress making the discussion of slavery taboo, and made slave-holding a prerequisite for holding office in South Carolina. Many Southerners very seriously believed that ending slavery would lead to an immediate race war and the extermination of the white race in the South. Others, North and South alike, did not believe that the United States could survive as a biracial or multiracial republic. As such, buying and freeing every slave would have to be only part of a much larger project to resettle freed slaves to Africa (since there could be no place for them in the white American West). Such complications would have made any mass purchase of enslaved people a complete non-starter in Washington.

For more on Southern views of the consequences of emancipation, see Alan Taylor, The Internal Enemy.

For more on the tumultuous political lead up to the war (and good counts of the war itself), see Allen Guelzo, Fateful Lightning and James McPherson, The Battle Cry of Freedom.

The value of the slave property in the United States in 1860 (3,953,761 slaves at $800 each) was $3.163 billion. From 1861-1865, the United States spent $3.090 billion on defense (though over 60% of that was in 1864/65). The United States government spent $78 million on everything in 1860.

So yes, theoretically the United States could have afforded to purchase all of the slaves in the South, but the cost was over 40 times the budget before the war, and only seemed like a good use of money in hindsight, when almost that money had been spent for the same result, with hundreds of thousands of men dying. It would have been politically impossible to push such a program through.

What Lincoln did to prevent war was to tell the South that they could keep their slaves - they could keep their slaves forever - but that they couldn't bring slaves into the territories. Slavery had to stay where it was, but it would not be interfered with there. What brought on the war was not any insistence by Lincoln that slavery be abolished, but rather his refusal to cede to the South's view that a state could unilaterally secede from the Union. By continuing to treat the South as being in the Union and resupplying the forts that the federal government still occupied, he forced the so-called Confederate States to start a war, lest they concede that they were still part of the Union through their passivity.