diff --git a/astropy/time/formats.py b/astropy/time/formats.py
index c83d6becbc2c..adc2ef33445d 100644
--- a/astropy/time/formats.py
+++ b/astropy/time/formats.py
@@ -121,6 +121,8 @@ class TimeFormat:
     ----------
     val1 : numpy ndarray, list, number, str, or bytes
         Values to initialize the time or times.  Bytes are decoded as ascii.
+        Quantities with time units are allowed for formats where the
+        interpretation is unambiguous.
     val2 : numpy ndarray, list, or number; optional
         Value(s) to initialize the time or times.  Only used for numerical
         input, to help preserve precision.
@@ -545,6 +547,7 @@ def to_value(self, jd1=None, jd2=None, parent=None, out_subfmt=None):
 class TimeJD(TimeNumeric):
     """
     Julian Date time format.
+
     This represents the number of days since the beginning of
     the Julian Period.
     For example, 2451544.5 in JD is midnight on January 1, 2000.
@@ -560,6 +563,7 @@ def set_jds(self, val1, val2):
 class TimeMJD(TimeNumeric):
     """
     Modified Julian Date time format.
+
     This represents the number of days since midnight on November 17, 1858.
     For example, 51544.0 in MJD is midnight on January 1, 2000.
     """
@@ -580,25 +584,39 @@ def to_value(self, **kwargs):
     value = property(to_value)
 
 
-class TimeDecimalYear(TimeNumeric):
-    """
-    Time as a decimal year, with integer values corresponding to midnight
-    of the first day of each year.  For example 2000.5 corresponds to the
-    ISO time '2000-07-02 00:00:00'.
+class TimeNumericNoQuantity(TimeNumeric):
     """
+    Numerical time format with a non-SI scale of time.
 
-    name = "decimalyear"
+    For these, quantities are not allowed, as the assumption that
+    one year always contains a fixed number of days does not hold.
+    """
 
     def _check_val_type(self, val1, val2):
         if hasattr(val1, "to") and hasattr(val1, "unit") and val1.unit is not None:
             raise ValueError(
-                "cannot use Quantities for 'decimalyear' format, as the unit of year "
-                "is defined as 365.25 days, while the length of the year is variable "
-                "in the 'decimalyear' format. Use float instead."
+                f"cannot use Quantities for {self.name!r} format, as the unit of year "
+                "is defined as 365.25 days, while the length of year is variable "
+                "in this format. Use float instead."
             )
         # if val2 is a Quantity, super() will raise a TypeError.
         return super()._check_val_type(val1, val2)
 
+
+class TimeDecimalYear(TimeNumericNoQuantity):
+    """
+    Time as a decimal year, with integer values corresponding to midnight
+    of the first day of each year.
+
+    For example 2000.5 corresponds to the ISO time '2000-07-02 00:00:00'.
+
+    Since for this format the length of the year varies between 365 and
+    366 days, it is not possible to use Quantity input, in which a year
+    is always 365.25 days.
+    """
+
+    name = "decimalyear"
+
     def set_jds(self, val1, val2):
         self._check_scale(self._scale)  # Validate scale.
 
@@ -657,7 +675,7 @@ def to_value(self, **kwargs):
 class TimeFromEpoch(TimeNumeric):
     """
     Base class for times that represent the interval from a particular
-    epoch as a floating point multiple of a unit time interval (e.g. seconds
+    epoch as a numerical multiple of a unit time interval (e.g. seconds
     or days).
     """
 
@@ -1962,7 +1980,7 @@ def value(self):
 
 class TimeEpochDate(TimeNumeric):
     """
-    Base class for support floating point Besselian and Julian epoch dates.
+    Base class for support of Besselian and Julian epoch dates.
     """
 
     _default_scale = "tt"  # As of astropy 3.2, this is no longer 'utc'.
@@ -1981,26 +1999,21 @@ def to_value(self, **kwargs):
     value = property(to_value)
 
 
-class TimeBesselianEpoch(TimeEpochDate):
-    """Besselian Epoch year as floating point value(s) like 1950.0."""
+class TimeBesselianEpoch(TimeEpochDate, TimeNumericNoQuantity):
+    """Besselian Epoch year as value(s) like 1950.0.
+
+    Since for this format the length of the year varies, input needs to
+    be floating point; it is not possible to use Quantity input, for
+    which a year always equals 365.25 days.
+    """
 
     name = "byear"
     epoch_to_jd = "epb2jd"
     jd_to_epoch = "epb"
 
-    def _check_val_type(self, val1, val2):
-        """Input value validation, typically overridden by derived classes."""
-        if hasattr(val1, "to") and hasattr(val1, "unit") and val1.unit is not None:
-            raise ValueError(
-                "Cannot use Quantities for 'byear' format, as the interpretation "
-                "would be ambiguous. Use float with Besselian year instead."
-            )
-        # FIXME: is val2 really okay here?
-        return super()._check_val_type(val1, val2)
-
 
 class TimeJulianEpoch(TimeEpochDate):
-    """Julian Epoch year as floating point value(s) like 2000.0."""
+    """Julian Epoch year as value(s) like 2000.0."""
 
     name = "jyear"
     unit = erfa.DJY  # 365.25, the Julian year, for conversion to quantities