How did people count before the invention of the number 0?

You don’t need zero to count quantities. Zero is the “nothing” value’s symbol.

You just go about naming every number, like one, two, three, etc., and there’s no limit to it. This is how it used to be in India, which is unsurprisingly the land where the modern placeholder number system was invented.

Here’s a small note on the numbering system, given in Sri Desiraju Hanumanta Rao’s translation of Sarga 38, Kishkindha Kanda, Valmiki Ramayana:¹

In the Indian counting system hundred thousands becomes one lakh (1,00,000), one hundred lakhs become one crore (1,00,00,000). The ancient Indian legionaries have names like

• आयुत (aayuta) = one thousand per unit
• शङ्कु (shanku) = one lakh crores, i.e., a trillion - 10¹²
• अर्बुद (arbuda) = 1000 × शङ्कु
• मध्यम (madhyama) = 10 × अर्बुद
• अन्तम् (antam) = 10 × मध्यम
• समुद्र (samudra) = 20 × अन्त्य
• परार्ध (paraardha) = 30 × समुद्र

Rama Tilaka says that by giving these numbers it is to be understood that ‘innumerable vaanaras are coming…’ But others hold the view that the ancients have organised military pattern hence particular nomenclature is given to each, apart from the generalisation of Rama Tilaka.

Govindaraja takes up the enumeration as given in Indian Astrology, which multiplies ten times each, as said in verse:

एकम् दश शतम् अस्मात् सहस्रम् अयुतम् ततः परम् लक्षम् ।
प्रयुतम् कोटिम् अथ अर्बुदम् वृन्दे खर्वम् निखर्वम् च ।
तस्मात् महा सरोजम् शन्कुम् सरिताम् पतिम् त्वत् अन्तम् ।
मध्यम् पराधम् आरूह्य अथ उत्तरम् दश गुणम् तथा ज्ञेयम् ॥

‘one, ten, hundred, thousand, ten thousand, one lakh, प्रयुतम् (prayutam), ten lakhs or a million, then a crore and then an अर्बुद (arbuda) ten-crore or a billion, and then ten arbuda-s makes one बृन्द (bRnda) and ten brinda-s make one खर्व (kharva) and ten kharva-s make one निखर्व (nikharva) and ten nikharva-s make one महा पद्म (mahaa padma) and ten mahaa padma-s make one शङ्कु (shanku), one lakh crores or a trillion, and ten shanku-s make one समुद्र (samudra) and then ten samudra-s make one अन्त (anta) and ten anta-s make a मध्यम (madhyama) and ten madhyama-s make one परार्ध (paraardha) it may be known thus…

By this, the nomenclature of million, billion and trillion was there in ancient days and this may be observed by the names: प्रयुतम् : ‘a million…’, अर्बुद : ‘a billion..’, and शङ्कु : ‘a trillion…’, and this exactly relates to the decimal system which is based on the number ten, in which the smaller units are related to the principal units as powers of ten (units, tens, hundreds, thousands, etc.).

Don’t mistake this as just having words for the powers of ten. In Sanskrit (and other Indian languages, but we’re talking about ancient times here) one can make up a word for any number by following certain rules and compounding words.

This way you essentially have a number system without numbers.