I wrote a short script to see the impact of file size depending on the protocol:
---
import shelve
import os.path

print("== Short value ==")
for proto in (0, 1, 2, 3, 4, 5):
    filename = 'shelve-picklev%s' % proto
    with shelve.open(filename, protocol=proto) as db:
        assert db._protocol == proto
        for x in range(1000):
            db[str(x)] = str(x)
    print(f'Protocol {proto}: {os.path.getsize(filename)} bytes')
    os.unlink(filename)
print()
print("== Large value ==")
large_value = [str(x) for x in range(1000)]
for proto in (0, 1, 2, 3, 4, 5):
    filename = 'shelve-picklev%s' % proto
    with shelve.open(filename, protocol=proto) as db:
        assert db._protocol == proto
        for x in range(10):
            db[str(x)] = large_value
    print(f'Protocol {proto}: {os.path.getsize(filename)} bytes')
    os.unlink(filename)
---

Output with Python 3.9.0b1 (on Fedora 32):
---
== Short value ==
Protocol 0: 90112 bytes
Protocol 1: 94208 bytes
Protocol 2: 94208 bytes
Protocol 3: 94208 bytes
Protocol 4: 94208 bytes
Protocol 5: 94208 bytes

== Large value ==
Protocol 0: 139264 bytes
Protocol 1: 139264 bytes
Protocol 2: 139264 bytes
Protocol 3: 139264 bytes
Protocol 4: 98304 bytes
Protocol 5: 98304 bytes
---

For short string values, protocol 0 produces smaller files than protocol 1 and higher.

For large value, protocol 4 and higher produce smaller files than protocol 3 and lower.