Suppose we build big machines to remove CO₂ from the atmosphere to get back down from 400 ppm back to pre-industrial levels. If we froze all that excess CO₂ into a big block of dry ice, how big would it be?
The atmosphere has a mass of about 5.15 × 10¹⁸ kg, and 0.414% of that is CO₂, by volume. The average molecular weight of dry air is about 28.95 g/mol, while CO₂ is 44.01 g/mol, making it 1.52 times the density of dry air. (The CO₂ page on WP says 1.53 times.) This means that CO₂ is about 0.629 / (0.629 + 100 - 0.414) = 0.628% of the atmosphere by mass, or 3.25 × 10¹⁶ kg. Pre-industrial levels were about 250 ppm, so we only need to remove about 164 ppm, or 164/414 of the total; that's about 1.29 × 10¹⁶ kg.
The density of dry ice varies from 1.4–1.6 g/cc, so this would be about 8–9 × 10¹² m³ of dry ice, roughly a cube 20 km on a side. This is the volume of about 50–60 trillion barrels of oil, almost a thousand times the size of the Ghawar supergiant oil field, the largest among the 40 000 or so oil fields in the world.
If you pump the CO₂ back into the ground, the high pressure will convert it to highly compressed supercritical CO₂, almost the same density as dry ice, so this is probably a reasonable estimate of how much underground storage space is needed.